What's the general solution for this differential equation?

Question

Dy/DX+3/x y=2x+5

Answer 1

dy/dx + (3/x)y = 2x+5

 

This is a first order linear differential equation.  

 

Multiply both sides by the integrating factor e^∫(3/x)dx = e^3lnx = x³ (for x>0)

 

So, (x³)(dy/dx) + (3x²)y = 2x⁴ + 5x³

 

                         (x³y)' = 2x⁴ + 5x³

 

Integrating both sides, we have x³y = (2/5)x⁵ + (5/4)x⁴ + C

 

                                                y = (2/5)x² + (5/4)x + Cx^-3 (x>0)