Hi there, I have a question on differential equations I dont understand, here it is:
find the general solution:
t (dx/dt) = x + (sin(x/t))^2
I think you have to use substitution for this one but dont know how, any ideas?
Define $\displaystyle u=\frac{x}{t}$. Then $\displaystyle \frac{du}{dt}=-\frac{x}{t^2}+\frac{1}{t}\frac{dx}{dt}$. Now, our original equation tells us $\displaystyle \frac{dx}{dt}=\frac{x}{t}+\frac{1}{t}\sin^2\frac{x }{t}$, and so:
$\displaystyle \frac{du}{dt}=-\frac{x}{t^2}+\frac{1}{t}\left(\frac{x}{t}+\frac{1 }{t}\sin^2\frac{x}{t}\right)$
$\displaystyle \frac{du}{dt}=\frac{1}{t^2}\sin^2\frac{x}{t}$
$\displaystyle \frac{du}{dt}=\frac{\sin^2{u}}{t^2}$
Which you should be able to solve for u(t). Then just use x=ut to find x.
--Kevin C.