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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
If $y = e^{a \sin–1} x, –1 < x < 1,$ then show that
$\big(1-\text{x}^2\big)\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{a}^2\text{y}=0\dot{}$
✓

Answer

$\frac{\text{dy}}{\text{dx}}=\text{e}^\text{a}\;\sin^{-1}\text{x}\frac{a}{\sqrt{1-\text{x}^2}}=\frac{\text{ay}}{\sqrt{1-\text{x}^2}}$
$\Rightarrow\sqrt{1-\text{x}^2}\ \dot{}\ \frac{\text{dy}}{\text{dx}}=\text{ay}.............(\text{i})$
$\Rightarrow\sqrt{1-\text{x}^2}\ \dot{}\ \frac{\text{d}^2\text{y}}{\text{dx}^2}-\frac{\text{x}}{\sqrt{1-\text{x}^2}} \dot{}\ \ \frac{\text{dy}}{\text{dx}}=\ \text{a} \frac{\text{dy}}{\text{dx}}$
$\Rightarrow\big(1-\text{x}^2\big)\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}\frac{\text{dy}}{\text{dx}}-\text{a}\sqrt{1-\text{x}^2}\frac{\text{dy}}{\text{dx}}=0$
$\Rightarrow\big(1-\text{x}^2\big)\frac{\text{d}^2\text{y}} {\text{dx}^2}-\text{x}\frac{\text{dy}}{\text{dx}}-\text{a}^2\text{y}=0 \ [\text{Using (i)}]$

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