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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
Find the particular solution of the differential equation $\frac{\text{dy}}{\text{dx}} - \text{3y} \cot \text{x} = \sin \text{2x}, $ given that y = 2 when $\text{x} = \frac{\pi}{2}.$
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Answer

Here, $\text{I.F.} = \text{e}^{\int - 3\cot {\text{x dx}}} = \frac{1}{\sin^{3}\text{x}}$
Solution is given by, $\text{y} \bigg(\frac{1}{\sin^{3} \text{x}}\bigg) = \int \frac{\sin \text{2x}}{\sin^{3} \text{x}} \text{dx} = 2 \int \frac{\cos \text{x}}{\sin^{2}\text{x}} \text{dx}$
$\Rightarrow \frac{\text{y}}{\sin^{3} \text{x}} = \frac{-2}{\sin \text{x}} + \text{c}$
$\text{when x} = \frac{\pi}{2}, \text{y} = 2 \Rightarrow \text{c = 4}$
$\therefore \frac{\text{y}}{\sin^{3} \text{x}} = \frac{-2}{\sin \text{x}} + 4 \text{ } \text{or } \text{y} = -2 \sin^{2} \text{x} + \text{ 4 } \sin^{3} \text{x}$

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