Differential equation $x^2\frac{dw}{dx} = \sqrt{w}(7x+5)$.

Question

Can someone steer me in the right direction to solve this DE?

$$x^2\dfrac{dw}{dx} = \sqrt{w}(7x+5)$$

So far, I arrived at: $$w^{-\frac12}dw=(7x+5)x^\frac12dx$$

Does this look right?

Answer 1

Hint: Write your equation in the form $$\frac{dw}{\sqrt{w}}=\frac{7x+5}{x^2}dx$$

Answer 2

After fixing the exponents,

$$w^{-1/2}w'=7x^{-1}+5x^{-2}$$ integrates in a straightforward way.

$$2\sqrt w=7\log x-5x^{-1}+C,\\w=\left(\frac72\log x-\frac 5{2x}+C\right)^2.$$