Differentialligninger - Matematik og naturfag i verdensklasse
Differentialligninger - Matematik og naturfag i verdensklasse
Differentialligninger - Matematik og naturfag i verdensklasse
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Analytiske løsninger<br />
b·f'(x)<br />
- —————————<br />
#11: 2<br />
a·f(x)<br />
Da f er en løsning til differentialligningen, er<br />
#12: f'(x) = f(x)·(b - a·f(x))<br />
Altså er<br />
b·f(x)·(b - a·f(x))<br />
g'(x) = - —————————————————————<br />
#13: 2<br />
a·f(x)<br />
der omformes til<br />
2<br />
b<br />
#14: g'(x) = b - ————————<br />
a·f(x)<br />
<strong>og</strong> dermed<br />
b ‚<br />
¦ ——— ¦<br />
#15: ¦ a ¦<br />
g'(x) = - b·¦-1 + ——————¦<br />
f(x) ƒ<br />
altså<br />
#16: g'(x) = - b·g(x)<br />
Funktionen g har derfor regneforskriften<br />
- b·x<br />
#17: g(x) = c·ê<br />
hvor c > 0 <strong>og</strong> dermed<br />
b<br />
———<br />
#18: a - b·x<br />
—————— - 1 = c·ê<br />
f(x)<br />
sættes<br />
#19: y = f(x)<br />
fås<br />
b<br />
———<br />
#20: a - b·x<br />
————— - 1 = c·ê<br />
y<br />
b ‚<br />
¦ ——— ¦<br />
#21: ¦ a - b·x ¦<br />
SOLVE¦————— - 1 = c·ê , y¦<br />
y ƒ<br />
b·x<br />
b·ê<br />
#22: y = ——————————————<br />
b·x<br />
a·(ê + c)<br />
altså<br />
36
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