Question

Solve the linear inhomogeneous differential equation using the constant variation method: y^,-2y/x=I +1/x

Answer 1

Corresponding homogeneous differential equation

$$y'-2y/x=0$$

$$\dfrac{dy}{y}=2\dfrac{dx}{x}$$

Integrate

$$\int\dfrac{dy}{y}=\int2\dfrac{dx}{x}$$

$$y=Cx^2$$

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

$$C'x^2+2xC-2\dfrac{Cx^2}{x}=1+\dfrac{1}{x}$$

$$C'=\dfrac{1}{x^2}+\dfrac{1}{x^3}$$

Integrate

$$C=-\dfrac{1}{x}-\dfrac{1}{2x^2}+c_1$$

Substitute

$$y=-x-\dfrac{1}{2}+c_1x^2$$