Thread: Use substitution to solve a differential equation

1. Use substitution to solve a differential equation

By using the substitution $v=x-y$, solve the differential equation $\frac{dy}{dx}+[1+(x-y)^2]cos^2x=sin^2x$, expressing $y$ in terms of $x$.

$\frac{dv}{dx}=1-\frac{dy}{dx}$

$1-\frac{dv}{dx}+[1+v^2]cos^2x=sin^2x$

How do I continue to remove the x in the trigo?

2. Re: Use substitution to solve a differential equation

Originally Posted by Punch
By using the substitution $v=x-y$, solve the differential equation $\frac{dy}{dx}+[1+(x-y)^2]cos^2x=sin^2x$, expressing $y$ in terms of $x$.

$\frac{dv}{dx}=1-\frac{dy}{dx}$

$1-\frac{dv}{dx}+[1+v^2]cos^2x=sin^2x$

How do I continue to remove the x in the trigo?
$\frac{dv}{dx}=1+\cos^2x+v^2\cos^2x-\sin^2x$

$=2\cos^2x+v^2\cos^2x$

$=\cos^2x(v^2+2)$

$\frac{dv}{v^2+2}=\cos^2xdx$

Can you proceed?

3. Re: Use substitution to solve a differential equation

Originally Posted by alexmahone
$\frac{dv}{dx}=1+\cos^2x+v^2\cos^2x-\sin^2x$

$=2\cos^2x+v^2\cos^2x$

$=\cos^2x(v^2+2)$

$\frac{dv}{v^2+2}=\cos^2xdx$

Can you proceed?
Yeap, thanks!