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Formation of a Differential Equation from a General Solution

Obtain the di...

Question

Obtain the differential equation from the relation $A x^{2} + B y^{2} = 1$ , where A and B are constants.

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Solution

$A x^{2} + B y^{2} = 1$
By differentiating, we get
$2 A x + 2 B y y_{1} = 0$
$A = - \frac{B y y_{1}}{x}$
$2 A x + 2 B y y_{1} = 0$
By differentiating, we get
$2 A + 2 B y y_{2} + 2 B y_{1}^{2} = 0$
$- B y y_{1} + x (B y y_{2} + B y_{1}^{2}) = 0$
$x y y_{2} + x (y_{1})^{2} - y y_{1} = 0$

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