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  • Solve the differential equation (x^{2}-1)frac{dy}{dx}+2xy=frac{1}{x^{2}-1}

Solve the differential equation (x^{2}-1)\frac{dy}{dx}+2xy=\frac{1}{x^{2}-1}

Answers (1)

  Given:

(x^{2}-1)\frac{dy}{dx}+2xy=\frac{1}{x^{2}-1}

To find: Solution of the given differential equation

       Rewriting the equations as,

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

It is a first order liner differential equation Compare it with,

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

        Calculate Integrating Factor,

IF=e^{\int p(x)dx}\\ \\ IF=e^{\int \frac{2x}{x^{2}-1}dx}\\ \\ \Rightarrow \int \frac{2x}{(x^{2}-1)}dx=\int \frac{(x+1)+(x-1)}{(x^{2}-1)}dx\\ \int \frac{dx}{x-1}+\int \frac{dx}{x+1}=ln\left ( x+1 \right )+ln\left ( x-1 \right )\\ Formula:\;\int \frac{dx}{x}=ln\;x\\ IF=e^{ln(x^{2}-1)} \\IF=x^{2}-1

Hence, solution of the differential equation is given by,

y.(IF)=\int q(x).(IF)dx\\ y(x^{2}-1)=\int \frac{1}{(x^{2}-1)^{2}}(x^{2}-1)dx\\ y(x^{2}-1)=\int \frac{1}{(x^{2}-1)}dx\\ Formula: \int \frac{1}{(x^{2}-1)}dx=\frac{1}{2}\log\left ( \frac{x-1}{x+1} \right )\\ \frac{1}{(x^{2}-1)}dx=\frac{1}{2}\log\left ( \frac{x-1}{x+1} \right )+c

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infoexpert24

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