9874321 skrev:Kan du vise steg-for-steg hvordan finne løsningskurven i oppgave 2?
Får bare et svar med imaginære tall og masse røtter..
[tex]\frac{\mathrm{d}x}{x^3} = t^2\mathrm{d}t \ \Rightarrow \ \int x^{-3}\mathrm{d}x = \int t^2\mathrm{d}t[/tex]
[tex]\frac{1}{-3+1}x^{-3+1} = \frac{1}{2+1}t^{2+1} + C \ \Rightarrow \ -\frac{1}{2x^2} = \frac{1}{3}t^3 + C[/tex]
[tex]\frac{1}{x^2} = -\frac{2}{3}t^3+C[/tex]
[tex]x^2 = \frac{1}{-\frac{2}{3}t^3+C} = \frac{\frac{3}{2}}{C-t^3}[/tex]
[tex]x(t) = \frac{\sqrt{\frac{3}{2}}}{\sqrt{C-t^3}}[/tex]
[tex]x(1) = 3 \ \Rightarrow \ \frac{\sqrt{\frac{3}{2}}}{\sqrt{C-1}} = 3[/tex]
[tex]3\sqrt{C-1} = \pm\sqrt{\frac{3}{2}} \ \Rightarrow \ C-1 = \frac{1}{9}\cdot\frac{3}{2}[/tex]
[tex]C = 1+\frac{1}{6} = \frac{7}{6}[/tex]
[tex]x(t) = \frac{\sqrt{3}}{\sqrt{2}\cdot\sqrt{\frac{7}{6}-t^3}} = \frac{\sqrt{3}}{\sqrt{\frac{2}{6}\left(7-6t^3\right)}} = \frac{3}{\sqrt{7-6t^3}}[/tex]