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Differential Equations — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDifferential Equations3 Marks
Question
For the following differntial equations verify that the accompanying function is a solution:
Differential equation Function
$\text{x}+\text{y}\frac{\text{dy}}{\text{dx}}=0$ $\text{y}=\pm\sqrt{\text{a}^2-\text{x}^2}$
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Answer

We have $\text{y}=\pm\sqrt{\text{a}^2-\text{x}^2}$ $\Rightarrow\text{y}^2=\text{a}^2-\text{x}^2\ ...(1)$ Given differential equation $\text{x}+\text{y}\frac{\text{dy}}{\text{dx}}=0$ Differentiating both sides of (1) with respect to x, we get $2\text{y}\frac{\text{dy}}{\text{dx}}=-2\text{x}$ $\Rightarrow\text{y}\frac{\text{dy}}{\text{dx}}={\text{x}}$ $\Rightarrow\text{x}+\text{y}\frac{\text{dy}}{\text{dx}}=0$Hence, the given function is the solution to the given differential equation.

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