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  • The differential equation ydy/dx +x =c represents: A. Family of hyperbolas B. Family of parabolas C. Family of ellipses D. Family of circles

The differential equation \mathrm{y} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{x}=\mathrm{c} represents:
A. Family of hyperbolas
B. Family of parabolas
C. Family of ellipses
D. Family of circles

Answers (1)

\\ y \frac{d y}{d x}+x=c$ \\$\Rightarrow y \frac{d y}{d x}=c-x$ \\$y d y=(c-x) d x$
integrate
\\y d y=(c-x) d x$ \\$\Rightarrow \int y d y=\int(c-x) d x$ \\$\Rightarrow \int y d y=\int c d x-\int x d x$ \\$\Rightarrow \frac{y^{2}}{2}=c x-\frac{x^{2}}{2}+k$
k is the integration constant
\\\Rightarrow \frac{y^{2}}{2}+\frac{x^{2}}{2}=c x+k$ \\$\Rightarrow \frac{y^{2}+x^{2}}{2}=c x+k$

This is the equation of circle because there is no ‘xy’ term and x^2 and y^2 have the same coefficient.

This equation represents the family of circles because for different values of c and k we will get different circles.

Option D is correct.

Posted by

infoexpert22

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