2. ordens line\303\246r differentialligning med konst. koefficenter (basic)
rev. 24.11.16restart:Vi betragter den f\303\270lgende differentialligning:dlign:=diff(x(t),t,t)-3*diff(x(t),t)+2*x(t)=cos(2*t);1. Unders\303\270gelse af den tilsvarende homogene ligning:Vi finder l\303\270sningen til den tilsvarende homogene ligning:dlign_hom:=diff(x(t),t,t)-3*diff(x(t),t)+2*x(t)=0;dsolve(dlign_hom,x(t));Af dette udtryk kan vi se at r\303\270dderne i karakterligningen er 1 og 2.Kontrol:solve(L^2-3*L+2,L);Som forventet!2. Unders\303\270gelse af den inhomogene ligning:Vi finder l\303\270sningen til den givne inhomogene l\303\270sning:dsolve(dlign,x(t));Det er her tydeligt at genkende LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlElTGhvbUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn som vi fandt ovenfor.Endvidere kan vi se at 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er en partikul\303\246r l\303\270sning.
Lad os tjekke dette ved at inds\303\246tte udtrykket i differentialligningens venstreside:x0:=-1/20*cos(2*t)-3/20*sin(2*t);er en partikul\303\246r l\303\270sning til LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEmZGxpZ25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw== .
Lad os tjekke dette ved at inds\303\246tte udtrykket i differentialligningens venstreside:diff(x0,t,t)-3*diff(x0,t)+2*x0;Som forventet!3. En \303\270nsket partikul\303\246r l\303\270sning:Vi \303\270nsker at finde den l\303\270sning til LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRUYvRjI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEmZGxpZ25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw== som opfylder de to begyndelsesbetingelser: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSNtbkdGJDYlUSIwRidGMi9GNlEnbm9ybWFsRidGMkZBRjJGQS1JI21vR0YkNi5RIj1GJ0YyRkEvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXRj1GMkZB og 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({dlign,x(0)=0,D(x)(0)=1},x(t));graf:=rhs(%);tangent:=t;plot({graf,tangent},t=-1..1,scaling=constrained,view=-1..2);