Question

a(xdy+2ydx)=xydy solve the differential equation​

Answer 1

Answer:

$\sf 2a\times log |x|=y-a\times log|y|+C$

Step-by-step explanation:

Given:

$\sf The\:differential\:equation\: a(x\:dy+2y\:dx)=xy\:dy$

To Find:

The solution for the given differential equation

Solution:

$\sf a(x\:dy+2y\:dx)=xy\:dy$

$\sf \implies ax\:dy+2ay\:dx=xy\:dy$

Dividing the whole equation by dx,

$\sf \implies ax\: \dfrac{dy}{dx} +2ay=xy\: \dfrac{dy}{dx}$

$\sf \implies 2ay=xy\: \dfrac{dy}{dx} -ax\: \dfrac{dy}{dx}$

Taking x dy/dx as common,

$\sf \implies 2ay=x\:\dfrac{dy}{dx} (y-a)$

Making it in the form of variable separable,

$\sf \implies \dfrac{2a}{x} =\dfrac{dy}{dx} \: \dfrac{y-a}{y}$

$\sf \implies \dfrac{2a}{x} \: dx=\dfrac{y-a}{y} \: dy$

$\sf \implies \dfrac{2a}{x} \: dx=\bigg(1-\dfrac{a}{y}\bigg) \: dy$

Integrating on both sides,

$\displaystyle \sf \int\limits {\dfrac{2a}{x} } \, dx =\int\limits {\bigg(1-\dfrac{a}{y}\bigg) } \, dy$

$\displaystyle \sf 2a \int\limits {\dfrac{1}{x} } \, dx =\int\limits {1} \, dy -a\int\limits {\dfrac{1}{y} } \, dy$

$\sf \implies 2a\times log |x|=y-a\times log|y|+C$

This is the solution of the given differential equation.

Answer 2

d^4y/dx^4-y=0 differential equations