[Math] Solve the following differential equation: $xy’ – y = x^2$

Question

I'm preparing to exam in Linear Algebra $2$ and I have problems with differential equations..

For example, the following exercise:

Solve the following differential equation: $xy' – y = x^2$.

I started to solve:

$$xy' – y = x^2$$

$$ \implies y' – \frac{y}{x} = x$$

I need to find some $u$ and multiply both sides by it:

$$uy' – \frac{u}{x}y = ux$$

I need somehow to satisfy the product rule of derivative, by finding $u$ such that $u' = -\frac{u}{x}$ and by this get: $(uy)' = uy' + u'y$.

I need the help to find $u$.

I need to find $u$ such that $u' = -\frac{u}{x}$.

How would you find $u$? thanks in advance.

Answer 1

Hint: $\dfrac{u'}{u}=(\ln u)'$, so $\dfrac{u'}{u}+\dfrac{1}{x}=0$ implies $(\ln u)'+(\ln x)'=0$.