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  • Find the general solution of (x + 2y^{3}) ^{frac{dy}{dx}=y}

Find the general solution of (x + 2y^{3}) {\frac{dy}{dx}=y}

Answers (1)

Given:

\left ( x+2y^{3} \right )\frac{dy}{dx}=y

To find: Solution of the differential equation

Rewriting the equation as

\frac{dx}{dy}=\frac{\left ( x+2y^{3} \right )}{y}\\\frac{dx}{dy}=\frac{x}{y}+2y^{2}\\ \frac{dx}{dy}-\frac{x}{y}=2y^{2}\\

It is a first order linear differential equation

Comparing it with

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

Calculation the integrating factor,

IF=e^{\int p(y)dy}\\ IF=e^{\int -\frac{1}{y}dy}\\ formula\; \frac{dt}{t}=\ln t\\ IF=e^{-\ln y}=\frac{1}{y}

Therefore, the solution of the differential equation is

x.\left ( IF \right )=\int q(y).(IF)dy\\ \frac{x}{y}=\int 2y \; dy\\ Formula: \int x^{n}dx=\frac{x^{n+1}}{n+1}\\ \frac{x}{y}=y^{2}+c\\ x=y^{3}+cy

Posted by

infoexpert24

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