1. Solve the ode

x^3y' + 2y = x^3 + 2x y(1) = e+1

Which method would you use to solve this?

2. Originally Posted by sarah232
x^3y' + 2y = x^3 + 2x y(1) = e+1

Which method would you use to solve this?
The equation is linear so first put it in standard form

$\displaystyle y' + \frac{2y}{x^3} = 1 + \frac{2}{x^2}$

The integrating factor is

$\displaystyle \mu = e^{-\frac{1}{x^2}}$ so

$\displaystyle \frac{d}{dx} e^{-\frac{1}{x^2}} \cdot y = \left( 1 + \frac{2}{x^2}\right) e^{-\frac{1}{x^2}}$ which integrates giving

$\displaystyle e^{-\frac{1}{x^2}} \cdot y = x e^{-\frac{1}{x^2}} + c$

I think you can take it from here.

3. Originally Posted by danny arrigo
The equation is linear so first put it in standard form

$\displaystyle y' + \frac{2y}{x^3} = 1 + \frac{2}{x^3}$

The integrating factor is

$\displaystyle \mu = e^{-\frac{1}{x^2}}$ so

$\displaystyle \frac{d}{dx} e^{-\frac{1}{x^2}} \cdot y = \left( 1 + \frac{2}{x^2}\right) e^{-\frac{1}{x^2}}$ which integrates giving

$\displaystyle e^{-\frac{1}{x^2}} \cdot y = x e^{-\frac{1}{x^2}} + c$

I think you can take it from here.
If $\displaystyle x^3\frac{dy}{dx} + 2y = x^3 + 2x$ then
$\displaystyle \frac{dy}{dx} + 2x^{-3}y = 1 + 2x^{-2}$.