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  • Form the differential equation by eliminating A and B in Ax^{2} + By^{2}= 1

Form the differential equation by eliminating A and B in Ax2 + By2 = 1

 

Answers (1)

Given :

 Ax2+By2=1

To find: Solution of the differential equation

Differentiate with respect to x

2Ax+2By\frac{dy}{dx}=0\\ Ax+By\frac{dy}{dx}=0....(i)\\ Formula: \frac{d}{dx}(x^{n})=nx^{n-1}\\ \frac{dy}{dx}=-\frac{Ax}{By}.....(ii)

Differentiate the curve (i) again to get,

A+B\left ( \frac{dy}{dx}\frac{dy}{dx}+y\frac{d^{2}y}{dx^{2}} \right )=0\\ -\frac{A}{B}=\left ( \left ( \frac{dy}{dx} \right )^{2}+y\frac{d^{2}y}{dx^{2}} \right )

Substituting this in eq(i)

\frac{dy}{dx}=-\frac{x}{y}\left ( \left ( \frac{dy}{dx} \right )^{2}+y\frac{d^{2}y}{dx^{2}} \right )\\ y\frac{dy}{dx}=-x \left ( \frac{dy}{dx} \right )^{2}-xy\frac{d^{2}y}{dx^{2}}\\ y\frac{dy}{dx}+x \left ( \frac{dy}{dx} \right )^{2}+xy\frac{d^{2}y}{dx^{2}}=0

Posted by

infoexpert24

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