Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDifferential Equations3 Marks
Question
Form the differential equation corresponding to $\text{y}^2=\text{a}(\text{b}-\text{x}^2)$ bt eliminating a and b.
✓
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)
$\text{y}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2=\frac{\text{y}}{\text{x}}\frac{\text{dy}}{\text{dx}}$
$\text{x}\Big\{\text{y}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big\}=\text{y}\frac{\text{dy}}{\text{dx}}$
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