Form the differential equation corresponding to y^2=a(b-x^2) bt eliminating a and b.

y^2=a(b-x^2) Differential it with respect to x, 2y dy dx =a(-2x) ...(1) Again, differential it with respect to x, 2 [y d^2y dx^2 + dy dx dy dx ]=-2a y d^2y dx^2 + ( dy dx )^2=- ( 2y -2x dy dx ) using equation (1) y d^2y dx^2 + ( dy dx )^2= y x dy dx x y d^2y dx^2 + ( dy dx )^2 =y dy dx

Form the differential equation corresponding to y^2=a(b-x^2) bt e…

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Question

Form the differential equation corresponding to $\text{y}^2=\text{a}(\text{b}-\text{x}^2)$ bt eliminating a and b.

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Answer

$\text{y}^2=\text{a}(\text{b}-\text{x}^2)$
Differential it with respect to x,
$2\text{y}\frac{\text{dy}}{\text{dx}}=\text{a}(-2\text{x}) ...(1)$
Again, differential it with respect to x,
$2\Big[\text{y}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\frac{\text{dy}}{\text{dx}}\times\frac{\text{dy}}{\text{dx}}\Big]=-2\text{a}$
$\text{y}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2=-\Big(\frac{2\text{y}}{-2\text{x}}\frac{\text{dy}}{\text{dx}}\Big)$
using equation (1)
$\text{y}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2=\frac{\text{y}}{\text{x}}\frac{\text{dy}}{\text{dx}}$
$\text{x}\Big\{\text{y}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big\}=\text{y}\frac{\text{dy}}{\text{dx}}$