Question

2dy/dx+y=x^2

Answer 1

$$2y'+y=x^2$$

$$y'+y/2=x^2/2$$

Integration factor

$$\mu(x)=e^{\int(1/2)dx}=e^{x/2}$$

$$e^{x/2}y'+\dfrac{1}{2}e^{x/2}y=\dfrac{1}{2}x^2e^{x/2}$$

$$\int d(e^{x/2}y)=\int \dfrac{1}{2}x^2e^{x/2}dx$$

$$\int \dfrac{1}{2}x^2e^{x/2}dx=x^2e^{x/2}-2\int xe^{x/2}dx$$

$$=x^2e^{x/2}-4xe^{x/2}+4\int e^{x/2}dx$$

$$=x^2e^{x/2}-4xe^{x/2}+8e^{x/2}+C$$

$$e^{x/2}y=x^2e^{x/2}-4xe^{x/2}+8e^{x/2}+C$$

$$y=x^2-4x+8+Ce^{-x/2}$$