Differential Equation with an Initial condition

[Zinggy]

Homework Statement

x(dy/dx) = 3y +x⁴cos(x), y(2pi)=0

Homework Equations

N/A

The Attempt at a Solution

I've tried a couple different ways to make this separable, but you always carry over a 1/dx or 1/dy term and I can never fully separate this. I've also tried to do a Bernoulli differential equation method by doing a change of variable and putting it in the form: xy'-3y = x⁴cos(x) but it's not quite the right format to allow that to work, I would need it to be y⁴cos(x) instead.

 

[Orodruin]

Hint: Integrating factors.

 

[Ray Vickson]

Homework Statement

x(dy/dx) = 3y +x⁴cos(x), y(2pi)=0

Homework Equations

N/A

The Attempt at a Solution

I've tried a couple different ways to make this separable, but you always carry over a 1/dx or 1/dy term and I can never fully separate this. I've also tried to do a Bernoulli differential equation method by doing a change of variable and putting it in the form: xy'-3y = x⁴cos(x) but it's not quite the right format to allow that to work, I would need it to be y⁴cos(x) instead.

Divide both sides of the DE by $x$. That gives a very standard first-order DE with a well-known solution. (Hint: the hint from #2).

 

[Miles123K]

Divide both sides by $x$ and rearrange into:
$\dot y -\frac 3 x y = x^3 cos(x)$
Since you are studying differential equations I trust that you can figure out how to solve this.