Answer to Question #284294 in Differential Equations for Aysu

Question

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

y^,-(2y/x)=-(3/x²)

Answer 1

Corresponding homogeneous differential equation

$$y'-\dfrac{2y}{x}=0$$

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

Integrate

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

$$y=Cx^2$$

Then

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

Substitute

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

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

$$C'=-\dfrac{3}{x^4}$$

Integrate

$$C=\int (-\dfrac{3}{x^4})dx$$

$$C=\dfrac{1}{x^3}+C_1$$

The general solution of the given differential equation is

$$y=\dfrac{1}{x}+C_1x^2$$