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  • Find the equation of a curve passing through origin and satisfying the differential equation (1+x^{2})frac{dy}{dx}+2xy=4x^{2}

Find the equation of a curve passing through origin and satisfying the differential equation (1+x^{2})\frac{dy}{dx}+2xy=4x^{2}

Answers (1)

Given:

\left ( 1+x^{2} \right )\frac{dy}{dx}+2xy=4x^{2} and (0,0) is a solution to the curve

 

To find: Equation of the curve satisfying the differential equation

Rewrite the given equation

\frac{dy}{dx}+\frac{2xy}{\left ( 1+x^{2} \right )}=\frac{4x^{2}}{1+x^{2}}

Comparing with

\frac{dy}{dx}+p(x)y=q(x)\\ p(x)=\frac{2x}{1+x^{2}}\\ q(x)=\frac{4x^{2}}{1+x^{2}}\\

Calculating Integrating Factor

IF=e^{\int p(x)dx}\\ IF=e^{\int \frac{2x}{1+x^{2}}dx}

Calculating

\int \frac{2x}{1+x^{2}}dx
Assume

1+x^{2}=t\\ 2x\; dx=dt\\ \\ \int \frac{dt}{t}=\ln(t)\\ Formula:\int \frac{dx}{x}=\ln x\\ Substituting \;t=1+x^{2}\\ \int \frac{2x}{1+x^{2}}dx=\ln(1+x^{2})\\ IF=e^{\ln(1+x^{2})}=(1+x^2)

Hence the solution is

y.(IF)=\int q(x).(IF)dx\\ y(1+x^{2})=\int \frac{4x^{2}}{1+x^{2}}(1+x^{2})dx\\ y(1+x^{2})=\frac{4}{3}x^{3}+c\\ Formula: \int x^{n}dx=\frac{x^{n+1}}{n+1}

Satisfying (0,0) in the equation of the curve to find the value of c

0+0=c

c=0

therefore equation of the curve is

y(1+x^{2})=\frac{4}{3}x^{3}\\ y=\frac{4x^{2}}{3(1+x^{2})}

 

Posted by

infoexpert24

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