Question #284294

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

y,-(2y/x)=-(3/x2)


1
Expert's answer
2022-01-05T17:40:29-0500

Corresponding homogeneous differential equation


y2yx=0y'-\dfrac{2y}{x}=0

dyy=2dxx\dfrac{dy}{y}=\dfrac{2dx}{x}

Integrate


dyy=2dxx\int \dfrac{dy}{y}=\int\dfrac{2dx}{x}

y=Cx2y=Cx^2

Then


y=Cx2+2Cxy'=C'x^2+2Cx

Substitute


Cx2+2Cx2Cx2x=3x2C'x^2+2Cx-\dfrac{2Cx^2}{x}=-\dfrac{3}{x^2}

Cx2=3x2C'x^2=-\dfrac{3}{x^2}

C=3x4C'=-\dfrac{3}{x^4}

Integrate


C=(3x4)dxC=\int (-\dfrac{3}{x^4})dx

C=1x3+C1C=\dfrac{1}{x^3}+C_1

The general solution of the given differential equation is


y=1x+C1x2y=\dfrac{1}{x}+C_1x^2


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