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<Text-field style="Text" layout="Normal"><Font bold="true" encoding="UTF-8" size="18">Kunsten at l\303\270se ligninger i Maple</Font><Font encoding="UTF-8">
\302\251</Font><Font italic="true"> Knud Nissen & Adept Scientific 2008</Font></Text-field>
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<Text-field style="Heading 2" layout="Heading 2"><Font italic="true" size="12">Indledning</Font></Text-field>Maple har en meget st\303\246rk kommando til at l\303\270se ligninger symbolsk: > LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmc29sdmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUShMaWduaW5nRidGL0YyL0YzUSdub3JtYWxGJ0Y9Rj0= > LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmc29sdmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUShsaWduaW5nRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSdVa2VuZHRGJ0YvRjJGQUZBRkE= > 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Den f\303\270rste form foruds\303\246tter, at alle de ubekendte fremg\303\245r af ligningen. Man kan alts\303\245 nemt l\303\270se en konkret ligning somLUkmc29sdmVHNiI2Iy8sJkkieEdGJCIiJCIiJiIiIiIiKA==Hvis man vil l\303\270se en symbolsk ligning p\303\245 denne form, s\303\245 g\303\245r det galtLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy8qJkkiYUdGJyIiIkkieEdGJ0YsSSJiR0YnMaple ved jo ikke, hvad der er den ubekendte. L\303\246g m\303\246rke til de kr\303\270llede parenteser. Det er Maples m\303\245de at h\303\245ndtere m\303\246ngder p\303\245: L\303\270sningen indeholder 3 elementer ( i dette tilf\303\246lde ligninger): 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og LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJhRidGL0YyLUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUYsNiVRInhGJ0YvRjJGOQ==. Det skal forst\303\245s som en parametrisering af l\303\270sningen, hvor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= og LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= kan antage vilk\303\245rlige v\303\246rdier, mens LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5er fastlagt som produktet af LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= og LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.I stedet m\303\245 man skriveLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8qJkkiYUdGJyIiIkkieEdGJ0YsSSJiR0YnRi0=Man kan ogs\303\245 f\303\245 skrevet l\303\270sning ud som en m\303\246ngde ved fx at tilf\303\270je m\303\246ngdeparenser omkring de ubekendte (det v\303\246re bekvemt i forskellige sammenh\303\246nge):LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8qJkkiYUdGJyIiIkkieEdGJ0YsSSJiR0YnPCNGLQ==Endelig kan man ogs\303\245 bruge solve-kommandoen til isolere et vilk\303\245rligt udtryk i en ligning, men Maple anbefaler ikke denne brug. FxLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8mSSJFR0YnNiNJJGtpbkdGJywkKiZJIm1HRiciIiJJInZHRiciIiMjRjFGMyokRjJGMw==Ligninger kan navngives, s\303\245 der senere kan refereres til dem via navnet (her eq1):PkkkZXExRzYiLywqKiRJInhHRiQiIiQiIiIqJEYoIiIjRipGKCEiJCEiI0YqIiIhL\303\270sningerne til ligningen kan ogs\303\245 navngives, og det er en uhyre nyttig facilitet, s\303\245 man senere kan f\303\245 fat i de enkelte l\303\270sninger:PkkiTEc2Ii1JJnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRJJGVxMUdGJEkieEdGJA==L\303\270sningerne returnes som en liste, og de enkelte elementer kan udtr\303\246kkes med LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRRj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GLDYjUSFGJ0Y+ eller LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=_. Fx er (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== indtastes som L_2):JkkiTEc2IjYjIiIjHvis du s\303\246tter kr\303\270llede parenteser om LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, f\303\245r du l\303\270sningerne p\303\245 formenPkkjTDJHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkSSRlcTFHRiQ8I0kieEdGJA==Dette kan v\303\246re meget nyttigt, hvis du skal substituere en af l\303\270sningerne i et udtryk. Fx kan l\303\270sningerne checkes s\303\245ledes:LUklc3Vic0clKnByb3RlY3RlZEc2JCZJI0wyRzYiNiMiIiNJJGVxMUdGKA==LUknZXhwYW5kRyUqcHJvdGVjdGVkRzYjLywqKiQsJiMiIiIiIiNGKyokIiImRipGKiIiJEYrKiRGKUYsRisjISIoRixGK0YtIyEiJEYsIiIh
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8" italic="true" size="12">S\303\245dan kan l\303\270sningsm\303\246ngden begr\303\246nses</Font></Text-field>Hvis du blot er interesseret i den positive l\303\270sning i ligningen LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlhGOkY+, kan du f\303\270je en begr\303\246nsningsulighed til ligningen:PkkjTDNHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkPCQtSSI+R0YoNiRJInhHRiQiIiEvKiRGLyIiI0YzRi8=Pr\303\270v ogs\303\245 at finde de negative l\303\270sninger til ligningen eq1 :LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDwkSSRlcTFHRicySSJ4R0YnIiIhRiw=Helt s\303\245 nemt g\303\245r det ikke altid, fx hvis vi kun er interesserede i en positiv l\303\270sning til ligningen:PkkkZXEyRzYiLywmKiRJInhHRiQiIiMiIiJGKEYqKiRJImFHRiRGKQ==LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JEkkZXEyR0YnSSJ4R0YnLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDwkSSRlcTJHRictSSI+R0YlNiRJInhHRiciIiFGLg==For at f\303\245 en positiv l\303\270sning ses det, at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. Dette kan vi klare med en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnYXNzdW1lRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw==-kommando:LUknYXNzdW1lRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMwSSJhR0YnIiIhLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDwkSSRlcTJHRictSSI+R0YlNiRJInhHRiciIiFGLg==Der er s\303\245 kommet en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEoJlRpbGRlO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDRi8= efter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Det betyder, at der er gjort antagelser om LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8" italic="true" size="12">S\303\245dan undg\303\245r du komplekse l\303\270sninger</Font></Text-field>L\303\270sning af ligningen LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRjs2JFEiMUYnRj5GPg== vil give 3 l\303\270sninger, hvoraf to er komplekse:LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8qJEkieEdGJyIiJCIiIkYrSom standard l\303\270ser solve-kommandoen over de komplekse tal. Dette kan du undg\303\245 ved at indl\303\246se pakken RealDomain. Som standard arbejder Maple over de komplekse tal - dette \303\246ndres til de reelle tal ved indl\303\246sning af RealDomain-pakken:QyQtSSV3aXRoRzYiNiNJK1JlYWxEb21haW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUhIiI=LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8tSSJeR0knOi01NDkzR1FFQzpcUHJvZ3JhbW1lclxNYXBsZX4xMi9saWJcbWFwbGUubWxhRic2JEkieEdGJyIiJCIiIkYvLUkndW53aXRoRzYiNiNJK1JlYWxEb21haW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ=
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8" italic="true" size="12">S\303\245dan f\303\245r du alle l\303\270sninger med</Font></Text-field>Ved l\303\270sning af ligningen 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, som har uendelig mange l\303\270sninger, returnerer Maple blot en enkelt:LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8tSSRzaW5HRic2I0kieEdGJyMiIiIiIiNGLQ==Hvis du vil have samtlige l\303\270sninger returneret, skal du s\303\246tte variablen LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVExX0VudkFsbFNvbHV0aW9uc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= til LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEldHJ1ZUYnLyUnaXRhbGljR0YuLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GMlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGS0Y4PkkxX0VudkFsbFNvbHV0aW9uc0c2IkkldHJ1ZUclKnByb3RlY3RlZEc=PkkjRkxHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkLy1JJHNpbkdGJzYjSSJ4R0YkIyIiIiIiI0YvResultatet skal forst\303\245s p\303\245 den m\303\245de, at _Z1~ er et vilk\303\245rligt helt tal, og variablen _B~ kan kun antage v\303\246rdierne 0 og 1. Tegnet ~ (tilde) efter _Z betyder, at Maple har gjort antagelser om _Z1 og _B1 (hvis du genberegner skifter navnene til _Z2 og _B2). Variablerne _B1~ og _Z1~ er lokale variabler. Dette har den konsekvens, at du ikke har direkte adgang til at tildele dem v\303\246rdier, men de kan dog fiskes ud fra FL ved brug af indets (= indeterminates):LUknaW5kZXRzRyUqcHJvdGVjdGVkRzYjSSNGTEc2Ig==M\303\246ngdeklammerne fjernes ved at bede om operanderne i udtrykket:LUkjb3BHJSpwcm90ZWN0ZWRHNiM8JEklX0IxfGlyRzYiSSVfWjF8aXJHRig=Naturligvis kan dette klares i \303\251n arbejdsgang, hvor vi desuden navngiver den fremkomne liste af 'ubestemte' med LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Dette sker for lette referencen til de 'ubestemte' (det er meget besv\303\246rligt at f\303\245 skrevet _Z1~):Pkkiekc2Ii1JI29wRyUqcHJvdGVjdGVkRzYjLUknaW5kZXRzR0YnNiNJI0ZMR0YkLUkmYWJvdXRHNiI2IyZJInpHRiQ2IyIiIg==LUkmYWJvdXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyZJInpHRic2IyIiIw==En specifik l\303\270sning findes s\303\245ledesLUklc3Vic0clKnByb3RlY3RlZEc2JDwkLyZJInpHNiI2IyIiIkYsLyZGKTYjIiIjIiIhSSNGTEdGKg==Lad os lige stille _EnvAllSolutions tilbage og rense variablen z:QyQ+STFfRW52QWxsU29sdXRpb25zRzYiLkYkISIiQyQ+SSJ6RzYiLkYkISIi
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8" italic="true" size="12">Ingen l\303\270sning</Font></Text-field>Har en ligning ingen l\303\270sninger, eller kan Maple ikke finde nogen, s\303\245 returneres ingenting: Mere pr\303\246cist den tomme udtryksf\303\270lge returneres (NULL).LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8pSSJlR0YnSSJ4R0YnIiIhRiw=Den n\303\246ste ligning har l\303\270sning, men Maple kan ikke finde en formel for den:LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8tSSRzaW5HRiQ2I0kieEdGJywmRi0iIiIhIiJGL0YtVed at s\303\246tte informationsniveauet for solve til 1 f\303\245s en advarsel, n\303\245r solve ikke kan finde en l\303\270sning:QyQ+JkkqaW5mb2xldmVsR0koX3N5c2xpYkc2IjYjSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0YmIiIiISIiLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8pSSJlR0YnSSJ4R0YnIiIhRiw=S\303\246t infolevel for solve tilbage til 0QyQ+JkkqaW5mb2xldmVsR0koX3N5c2xpYkc2IjYjSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0YmIiIhISIi
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8" italic="true" size="12">Numerisk l\303\270sning</Font></Text-field>Vi betragter nu en ligning, om hvilken det vides, at l\303\270sningerne ikke alle kan udtrykkes ved de til r\303\245dighed v\303\246rende funktionsudtryk:PkkkZXEzRzYiLywkLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJInhHRiQiIiNGLQ==LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JEkkZXEzR0YnSSJ4R0YnVi fik et svar, men dette ligner n\303\246rmest en omformulering af sp\303\270rgsm\303\245let. Vi m\303\245 derfor l\303\270se ligningen numerisk.Der er to former fuldst\303\246ndig som for solve. LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJGVxM0dGJ0kieEdGJw==LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJGVxM0dGJzwjSSJ4R0YnDet bem\303\246rkes, at vi kun f\303\245r \303\251n l\303\270sning. De andre kan f\303\245s ved enten at angive et s\303\270geinterval eller et startg\303\246t. Igen kan de to forskellige former med og uden { } bruges. F\303\270rst pr\303\270ver vi at angive et s\303\270geinterval:LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJGVxM0dGJy9JInhHRic7IiIiIiIjLUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJGVxM0dGJzwjL0kieEdGJzshIiMiIiI=Her pr\303\270ver vi med et startg\303\246t:LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJGVxM0dGJy9JInhHRiciIiM=LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJGVxM0dGJzwjL0kieEdGJyEiJA==Man kan fort\303\246lle Maple, at den skal undg\303\245 en l\303\270sning, fx hvis der (som her) er tale om en triviel l\303\270sning, som vi ikke beh\303\270ver Maple til at finde:LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiVJJGVxM0dGJ0kieEdGJy9JJmF2b2lkR0YnPCMvRioiIiE=fsolve returnerer uevalueret, n\303\245r den ikke kan finde en l\303\270sning: LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJGVxM0dGJy9JInhHRic7IiIjIiIkDet er en god ide at plotte i forbindelse med l\303\270sning af ligninger, fx kunne man plotte venstre- og h\303\270jresiden af eq2. Maple kommandoerne rhs og lhs udtr\303\246kker hhv. h\303\270je- og venstre side af en ligning:LUklcGxvdEc2IjYkPCQtSSRsaHNHJSpwcm90ZWN0ZWRHNiNJJGVxM0dGJC1JJHJoc0dGKUYqL0kieEdGJDshIiQiIiQ=fsolve accepterer f\303\270lgende varianter med s\303\270geinterval foruden de tilsvarende med kr\303\270llede parenteser:LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJHNpbkdGJDsiIiUiIig=LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSRzaW5HRiQ2I0kieEdGJy9GLDsiIiUiIig=LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQvLUkkc2luR0YkNiNJInhHRiciIiEvRi07IiIlIiIoLUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUtSSRzaW5HRiQ2I0kieEdGJ0YsOyIiJSIiKA==
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8" italic="true" size="12">Lambert kommer p\303\245 banen</Font></Text-field>Maples evne til at l\303\270se ligninger er ganske omfattende, og der kan sagtens dukke overraskelser op i form af funktioner, man aldrig har h\303\270rt om. Lad os se p\303\245 ligningenLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRiw2JUY6LUYjNiRGLkY+RkBGPg==.Almindelig folklore siger, at denne ligning indeholder transcendente funktioner LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSVtc3VwR0YkNiUtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjhRJ2l0YWxpY0YnRjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjdGN0Y3 p\303\245 en ikke-algebraisk m\303\245de, og at sammenblandingen af algebraiske funktioner LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSVtc3VwR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIyRicvRjtRJ25vcm1hbEYnRkMvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkNGQ0ZD og transcendente funktioner LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSVtc3VwR0YkNiUtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjhRJ2l0YWxpY0YnRjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjdGN0Y3 ikke tillader en symbolsk l\303\270sning. Traditionelt vil man derfor l\303\270se ligningen numerisk (vha fsolve).Grafen afsl\303\270rer, at der er 3 reelle l\303\270sninger til ligningenLUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmPCQpIiIjSSJ4R0YnKiRGLEYrL0YsOyEiJiIiJi9JInlHRic7RjAiIz8vSShzY2FsaW5nR0YnSS51bmNvbnN0cmFpbmVkR0YnDer er naturligvis de to ret trivielle l\303\270sninger LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk= og LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjRGJ0Y5Rjk=, og en tredje mellem -1 og 0:LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQvKiRJInhHRiciIiMpRixGKy9GKzshIiIiIiE=Men symbolsk regning kan ogs\303\245 lade sig g\303\270reLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8qJEkieEdGJyIiIylGLEYrRis=Helt fint, men hvad er LambertW for en st\303\270rrrelse? I f\303\270lge Maples hj\303\246lpetekster er LambertW den omvendte funktion til 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. Et blik p\303\245 grafen for 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 afsl\303\270rer, at funktionen ikke er injektivLUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlKiZJInhHRiciIiItSSRleHBHRiQ2I0YqRisvRio7ISIlIiIlL0kieUdGJzshIiMiIik=Minimum befinder sig i LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1JI21uR0YkNiRRIjFGJ0Y5Rjk=, og minimumsv\303\246rdien er LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSVtc3VwR0YkNiUtSSNtaUdGJDYlUSJlRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiVGKy1JI21uR0YkNiRRIjFGJ0YvRi8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUZKNiNRIUYnRi8=. N\303\245r vi laver den omvendte, er vi derfor n\303\270dt til at dele op i to: Grenen gennem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMEY0RjRGNA==, der ved omvending giver LambertW-funktionen, og den anden gren (i 3. kvadrant), der ved omvending giver den s\303\245kaldte LambertW(-1,x) funktion. Definitions- og v\303\246rdim\303\246ngder for de to funktioner er s\303\245ledes: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 betyder, at ligningen 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 har l\303\270sningenLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSlMYW1iZXJ0V0YnL0YwRj1GOS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjJGOUY5Rjk= hvis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJiZnZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOUY5og l\303\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 hvis 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.Hvis du vil studre LambertW funktionerne n\303\246rmere, kan du f\303\245 Maples procedure til beregning skrevet ud i detaljer. G\303\270r s\303\245ledes fjern kolon efter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElZXZhbEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRKUxhbWJlcnRXRicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnRj9GPy1GLDYjUSFGJ0Y/ inden du udf\303\270rer kommandoen.QyQtSSppbnRlcmZhY2VHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JLHZlcmJvc2Vwcm9jR0YoIiIkISIiQyQtSSVldmFsRyUqcHJvdGVjdGVkRzYjSSlMYW1iZXJ0V0c2JEYlSShfc3lzbGliRzYiISIi
<Text-field style="Heading 2" layout="Heading 2"><Font italic="true" size="12">Flere ligninger</Font></Text-field>Skal du l\303\270se et system best\303\245ende af flere ligninger med flere ubekendte, skal du blot bundte ligningerne med {} og tilsvarende med variablerne. FxLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDwkLywmSSJ4R0YnIiIiSSJ5R0YnIiIjIiIkLywmRi5GLSokRiwhIiJGLUYtPCRGLEYuDen numeriske ligningsl\303\270ser fsolve l\303\270ser ogs\303\245 ligningssystemer. Der er tre varianter af syntaksen:PkkkZXE0RzYiPCQvLCZJInhHRiQiIiNJInlHRiQiIiIiIiEvLCYqJEYoRikiIiQqJEYqIiIlISIiRiw=LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJGVxNEdGJzwkSSJ4R0YnSSJ5R0YnMed startg\303\246t:LUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJGVxNEdGJzwkL0kieEdGJyIiIS9JInlHRiciIiI=Med s\303\270geintervallerLUknZnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJJGVxNEdGJzwkL0kieEdGJzsiIiEiIiIvSSJ5R0YnOyEiI0YuDet er naturligvis en god id\303\251 at bruge implicitplot f\303\270rst, s\303\245 man kan se, hvor der er sk\303\246ringspunkter:LSZJJnBsb3RzRzYiNiNJLWltcGxpY2l0cGxvdEdGJTYmSSRlcTRHRiUvSSJ4R0YlOyEiIiIiIi9JInlHRiVGLC9JKm51bXBvaW50c0dGJSImKysi
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8" italic="true" size="12">Implicitte l\303\270sninger</Font></Text-field>Til tider vil du se l\303\270sninger returneret implicit i termer af RootOf( ), som fx i ligningssystemet:LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDwkLywmKiZJInhHRiciIiNJInlHRidGLiIiIiEiI0YwIiIhLywoKiRGLUYuRjAqJEYvRi5GMCEiJUYwRjI3JEYtRi8=For at f\303\245 alle l\303\270sninger frem, benyttes kommandoen allvalues:LUkqYWxsdmFsdWVzRzYiNiM3IzckL0kieEdGJC1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCgqJEkjX1pHRiwiIiMiIiIhIiVGNCokLUYrNiMsKEYxRjUqJEYyIiIlRjRGM0Y0RjNGNC9JInlHRiRGNw==Hvis du vil undg\303\245 alt det RootOf( ) stads, kan du s\303\246tte variablen LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEtX0VudkV4cGxpY2l0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== til LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEldHJ1ZUYnLyUnaXRhbGljR0YuLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GMlEnbm9ybWFsRic=.Ogs\303\245 her er det en god ide at checke grafisk, at alle l\303\270sninger er med:LSZJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJLWltcGxpY2l0cGxvdEdGKDYmPCQvLCYqJkkieEdGKCIiI0kieUdGKEYxIiIiISIjRjMiIiEvLCgqJEYwRjFGMyokRjJGMUYzISIlRjNGNS9GMDshIiYiIiYvRjJGPC9JKm51bXBvaW50c0dGKCIlKz8=JSFH
<Text-field style="Heading 2" layout="Heading 2"><Font italic="true" size="12">Uligheder</Font></Text-field>Vi har allerede set, hvordan vi kan benytte en ulighed til at begr\303\246nse l\303\270sningsm\303\246ngden. Alts\303\245 fxPkkkZXE1RzYiLywqKiRJInhHRiQiIiRGKSokRigiIiMiIiJGKEYsRilGLCIiIQ==LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JEkkZXE1R0YnSSJ4R0YnLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDwkSSRlcTVHRicySSJ4R0YnIiIhRiw=Almindelige uligheder h\303\245ndteres s\303\245ledes:LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1JIj5HRiU2JCwoKiRJInhHRiciIiMiIiJGLiEiJEYvRjAiIiFGLg==Hvor resultatet skal tolkes s\303\245ledes: 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.LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1JIjxHRiU2JCwoKiRJInhHRiciIiMiIiJGLiEiJEYvRjAiIiFGLg==
<Text-field style="Heading 2" layout="Heading 2"><Font italic="true" size="12">Manipulation af ligninger</Font></Text-field>Ligninger kan navngives og manipuleres som andre objekter. Maple underst\303\270tter line\303\246re regninger med ligninger, dvs., de kan l\303\246gges sammen og tr\303\246kkes fra hinanden ligesom man kan gange og dividere ligninger med skalarer.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kan l\303\270ses i fuld detalje med de line\303\246re operationerQyY+SSRlcTFHNiIvLCZJInhHRiUiIiNJInlHRiUiIiQiIikhIiI+SSRlcTJHRiUvLCZGKCIiJUYqRi1GKUYtPkkkZXEzRzYiLCZJJGVxMUdGJCIiI0kkZXEyR0YkISIiL0kieUc2Ii1JJnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRJJGVxM0dGJEYjPkkkZXE0RzYiLCZJJGVxMUdGJCIiIkkkZXEyR0YkIiIkL0kieEc2Ii1JJnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRJJGVxNEdGJEYjAlternativt kan den sidste del l\303\270ses ved at substituere LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk= i fx LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkZXEyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw==, og l\303\270se mht. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEY5PkkkZXE0RzYiLUklc3Vic0clKnByb3RlY3RlZEc2JC9JInlHRiQiIiNJJGVxMkdGJA==L0kieEc2Ii1JJnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRJJGVxNEdGJEYjNaturligvis kan ligningssystemet ogs\303\245 l\303\270ses i \303\251t hugLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDwkSSRlcTFHRidJJGVxMkdGJzwkSSJ4R0YnSSJ5R0YnVed mere komplicerede operationer er der et par gode teknikker: Venstresiden og h\303\270jresiden af ligningen kan udskilles vha kommandoerne > lhs(Ligning) > rhs(Ligning)hvor lhs st\303\245r for left-hand-side og rhs for right-hand-side. Herefter kan den samme operation p\303\245 begge sider af lighedstegnet. Hvis du fx vil kvadrere, skrives det s\303\245ledes > LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVElZXFTcUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJGxoc0YnRi9GMi1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRI2VxRidGL0YyRjlGOS1GIzYkLUkjbW5HRiQ2JFEiMkYnRjlGOS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZkby1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZgby1GLDYlUSRyaHNGJ0YvRjJGUkY5Den anden teknik er en generel Maple teknik, hvor den universelle operator map benyttes til at udf\303\270re samme operation p\303\245 begge sider af lighedstegnet. Hvis du fx vil kvadrere, skrives det s\303\245ledes > 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 gives et par eksempler p\303\245 de to teknikker til l\303\270sning af ligningen 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\303\270rst med lhs og rhs:PkkjZXFHNiIvLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCZJInhHRiQiIiIhIiNGLiwmRi1GLiEiJUYuPkkkZXExRzYiLyokLUkkbGhzRyUqcHJvdGVjdGVkRzYjSSNlcUdGJCIiIyokLUkkcmhzR0YpRipGLA==PkkkZXEyRzYiLywmLUkkbGhzRyUqcHJvdGVjdGVkRzYjSSRlcTFHRiQiIiItSSRyaHNHRilGKiEiIiIiIQ==PkkkZXEzRzYiLUknZXhwYW5kRyUqcHJvdGVjdGVkRzYjSSRlcTJHRiQ=PkkiTEc2Ii1JJnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRJJGVxM0dGJDwjSSJ4R0YkDern\303\246st med mapPkkjZXFHNiIvLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCZJInhHRiQiIiIhIiNGLiwmRi1GLiEiJUYuPkkkZXExRzYiLUkkbWFwRyUqcHJvdGVjdGVkRzYkZio2I0kiekdGJEYkNiRJKW9wZXJhdG9yR0YkSSZhcnJvd0dGJEYkKiQ5JCIiI0YkRiRGJEkjZXFHRiQ=etc.Da vi selv har pillet ved ligningerne, kan der v\303\246re sluppet fremmedr\303\270dder ind i ligningen. Det kan unders\303\270ges ved at s\303\246tte l\303\270sningerne ind i den oprindelige ligning:LUklc3Vic0clKnByb3RlY3RlZEc2JCZJIkxHNiI2IyIiIkkjZXFHRig=LUklc3Vic0clKnByb3RlY3RlZEc2JCZJIkxHNiI2IyIiI0kjZXFHRig=LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy8qJCIiJSMiIiIiIiNGLg==
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8" italic="true" size="12">\303\230velser</Font></Text-field>L\303\270s ligningssystemet, som g\303\246lder for vilk\303\245rlige trekanter (arealformel og cosinusrelationen), men ikke med hensyn til noget bestemt - dvs. 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:QyQ+SSRlcTFHNiIvSSJUR0YlLCQqKEkiYUdGJSIiIkkiYkdGJUYrLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJIkNHRiVGKyNGKyIiIyEiIg==QyQ+SSRlcTJHNiIvKiRJImNHRiUiIiMsKCokSSJhR0YlRikiIiIqJEkiYkdGJUYpRi0qKEYsRi1GL0YtLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJIkNHRiVGLSEiIyEiIg==Maple leder nu efter alle mulige relationer mellem de indg\303\245ende st\303\270rrelser. Den finder to almene relationer. I det ene - ikke trivielle - tilf\303\246lde slipper Maple af med arealet T, og finder herved den omvendte cosinusrelation. I det andet - ikke trivielle - tilf\303\246lde slipper Maple af med C, og finder herved en version af Herons formel.Kvadrer denne l\303\270sning, og faktoriser det fremkomne resultat med factor. Hvad skal der til f\303\270r Herons formel st\303\245r der?