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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
For the following differntial equations verify that the accompanying function is a solution:
Differential equation Function
$\text{y}=\Big(\frac{\text{dy}}{\text{dx}}\Big)^2$ $\text{y}=\frac{1}{4}(\text{x}\pm\text{a})^2$
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Answer

We have
$\text{y}=\frac{1}{4}(\text{x}\pm\text{a})^2\ ...(1)$
Differentiating both sides of (1) with respect to x, we get
$\frac{\text{dy}}{\text{dx}}=\frac{1}{4}\times2(\text{x}\pm\text{a})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{1}{2}(\text{x}\pm\text{a})$
Squaring both sides we get
$\Rightarrow\Big(\frac{\text{dy}}{\text{dx}}\Big) ^2=\Big[\frac{1}{2}(\text{x}\pm\text{a})\Big]^2$
$\Rightarrow\Big(\frac{\text{d}\text{y}}{\text{dx}}\Big)=\frac{1}{4}(\text{x}\pm\text{a})^2$
$\Rightarrow\Big(\frac{\text{d}\text{y}}{\text{dx}}\Big)^2=\text{y}$
$\therefore\text{y}=\Big(\frac{\text{dy}}{\text{dx}}\Big)^2$
Hence, the given function is the solution to the given differential equation.

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