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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
For the following differntial equations verify that the accompanying function is a solution:
Differential equation Function
$\text{x}+\text{y}\frac{\text{dy}}{\text{dx}}=0$ $\text{y}=\pm\sqrt{\text{a}^2-\text{x}^2}$
✓

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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