Question

Solve differential equation xy'+2y= -x^3+x, \ y(1)=2

First order differential equations
ANSWERED
asked 2021-05-01
Solve differential equation \(xy'+2y= -x^3+x, \ y(1)=2\)

Answers (1)

2021-05-02

Divide both sides by x
\(y'+ \frac{2}{x}y= -x^2+1\)
This is first order linear differential equation of form
\(p(x)= \frac{2}{x}, \ q(x)= -x^2+1\)
Integrating factor is
\(=e^{\int p(x)dx}\)
\(=e^{\int \frac{2}{x}dx}\)
\(=e^{2 logx}\)
\(=e^{log x^2}\)
\(=x^2\)
equation is
\(y \cdot (I.F.)= \int (I.F.) q(x)dx+c\)
\(yx^2= \int x^2(-x+1)dx+c\)
\(yx^2=\int-x^4+x^2dx+c\)
\(x^2y= -\frac{x^5}{5}+\frac{x^3}{3}+c\)
\(y= -\frac{x^3}{5}+\frac{x}{3}+cx^{-2}\)
Given initial condition
\(y(1)=2\)
\(2= -\frac{1}{5}+\frac{1}{3}+\frac{c}{1}\)
\(2= \frac{-3+5}{15}+c\)
\(2= \frac{2}{15}+c\)
\(2- \frac{2}{15}=c\)
\(\frac{30-2}{15}=c\)
\(\frac{28}{15}=c\)
\(y= -\frac{x^3}{5}+\frac{x}{3}+\frac{28}{15x^2}\)

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