The difference between degree and order

Question:

The difference between degree and order of differential equation that represents the family of curves given by $y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right), a>0$ is

Solution:

$y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right)$

$2 y y^{\prime}=a$

$y^{2}=2 y y^{\prime}\left(x+\frac{\sqrt{2 y y^{\prime}}}{2}\right)$

$y=2 y^{\prime}\left(x+\frac{\sqrt{y y^{\prime}}}{\sqrt{2}}\right)$

$y-2 x y^{\prime}=\sqrt{2} y^{\prime} \sqrt{y y^{\prime}}$

$\left(y-2 x \frac{d y}{d x}\right)^{2}=2 y\left(\frac{d y}{d x}\right)^{3}$

$D=3 \& O=1$

$D-O=3-1=2$

 

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