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Q.
The differential equation representing the family of curves $y^2=2c (x+ \sqrt c)$ where $c$ is a positive parameter, is of
AIEEEAIEEE 1999Differential Equations
Solution:
Given, $ y^2=2c(x+ \sqrt c) ...(i)$
On differentiating w.r.t. $x$, we get
$ 2y \frac {dy}{dx}=2c \Rightarrow c=y \frac {dy}{dx}$
On putting this value of $c$ in Eq. (i), we get
$ y^2=2y \frac {dy}{dx} \bigg (x+ \sqrt {y \frac {dy}{dx}}\bigg )$
$\Rightarrow y=2 \frac {dy}{dx}.x+2y^{1/2} \bigg (\frac {dy}{dx} \bigg )^{3/2}$
$\Rightarrow y-2x \frac {dy}{dx}= 2 \sqrt y \bigg (\frac {dy}{dx} \bigg )^{3/2}$
$\Rightarrow \bigg (y-2x \frac {dy}{dx}\bigg )^2=4y \bigg (\frac {dy}{dx}\bigg )^3$
Therefore, order of this differential equation is $1$ and degree is $3$.