Which of the following ODE's is linear and how do I solve it?

Question

(1/x^2)(dy/dx)+4(y/x)=x

and

y(dy/dx)+4(y/x)=x

I also need to put the equation into the standard form

Answer 1

A linear first-order DE has the form

dy/dx + f(x)y = g(x)

(1/x²)(dy/dx) + 4(y/x) = x

This can be written as:

dy/dx + 4xy = x³

This is a linear differential equation.

The Integrating factor is

e^∫f(x)dx = e^∫4xdx = e^(2x²)

e^(2x²).y = ∫x³e^2x²dx

∫x³e^2x²dx

u = 2x²

du = 4xdx

∫x³e^2x²dx

= (1/8)∫u(e^u)du

Integrate this by parts to get:

(1/8)u(e^u) - (1/8)∫e^udu

= (1/8)u(e^u) - (1/8)e^u + c

= (1/8)(2x²)e^(2x²) - (1/8)e^(2x²) + c

(e^2x²)y = (1/4)x²e^(2x²) - (1/8)e^(2x²) + c

y = (1/4)x² - (1/8) + c.e^(-2x²)

Answer 2

i think 2nd equation is linear, just look at its degree, 1st order..