Question

Solve the first order linear inhomogeneous differential equation using the constant variation method

y^,- (3y/x)=x

Answer 1

Corresponding homogeneous differential equation

$$y'-\dfrac{3y}{x}=0$$ $$\dfrac{dy}{y}=3\dfrac{dx}{x}$$

Integrate

$$\int \dfrac{dy}{y}=\int 3\dfrac{dx}{x}$$ $$\ln (|y|)=3\ln (|x|)+\ln C$$ $$y=Cx^3$$

The general solution of the homogeneous differential equation is

$$y_h=Cx^3$$

Use the constant variation method

$$y'=C'x^3+3x^2 C$$

Substitute

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

Integrate

$$C=\int x^{-2}dx$$ $$C=-\dfrac{1}{x}+C_1$$

Finf the general solution of the non homogeneous differential equation

$$y=(-\dfrac{1}{x}+C_1)x^3$$ $$y=-x^2+C_1x^3$$