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Q.
The differential equation of all parabolas each of which has a latus rectum 4a and whose axes are parallel to the x-axis is of order 2 degree_________
Differential Equations
Solution:
Equation to the family of parabolas is $(y-k)^{2}=4a(x-h)$
$2(y-k)\frac{dy}{dx}=4a$ (differentiating w.r.t. $x$)
or $(y-k)\frac{dy}{dx}=2a\,... (1)$
or $\left(y-k\right) \frac{d^{2}y}{dx^{2}}+\left(\frac{dy}{dx}\right)^{2}=0$ (differentiating w.r.t. $x$)
or $2a \frac{d^{2}y}{dx^{2}}+\left(\frac{dy}{dx}\right)^{3}=0$ [substituting $y - k$ from equation (1)]
Hence, the order is $2$ and the degree is $1$