Answer to Question #350103 in Differential Equations for chand

Question

xdy/dx = x² +5y

Answer 1

$$xdy/dx=x^2+5y$$

$$y'-\dfrac{5}{x}y=x$$

Integrating factor

$$IF=e^{-\int (5/x)dx}=x^{-5}$$

$$x^{-5}y'-5x^{-6}y=x^{-4}$$

$$d(x^{-5}y)=x^{-4}dx$$

Integrate

$$\int d(x^{-5}y)=\int x^{-4}dx$$

$$x^{-5}y=-\dfrac{1}{3}x^{-3}+C$$

$$y=-\dfrac{1}{3}x^2+Cx^5$$