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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
Find a particular solution of the differential equation $\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}=4\text{x}\ \text{cosec}\ \text{x}\ (\text{x}\neq0),$ $\text{given that y}=0\ \text{when x}=\frac{\pi}{2}.$
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Answer

Given: Differential equation $\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}=4\text{x}\ \text{cosec}\ \text{x}$
Comparing this equation with $\frac{​\text{dy}​}{\text{dx}}+\text{Py}=\text{Q},\ \ \text{P}=\cot\text{x}\ \text{and}\ \text{Q}=4\text{x}\ \text{cosec}\ \text{x}$
$\int\text{P}\ \text{dx}=\int\cot\text{x}\ \text{dx}=\log\sin\text{x}$ $\text{I.F}=\text{e}^{\int\text{p dx}}=\text{e}^{\log\sin\text{x}}=\sin\text{x}$
The general solution is $\text{y}(\text{I.F})=\int\text{Q}(\text{I.F}.)\ \text{dx}+\text{c}$
$\Rightarrow\ \ \text{y}(\sin\text{x})=\int4\text{x}\ \text{cosec}\ \text{x}\ \sin\text{x}\ \text{dx}+\text{c}$ $\Rightarrow\ \ \text{y}(\sin\text{x})=4\int\text{x}.\frac{1}{\sin\text{x}}\sin\text{x}\ \text{dx}+\text{c}$
$\Rightarrow\ \ \text{y}(\sin\text{x})=4\int\text{x}\ \text{dx}+\text{c}=4\frac{\text{x}^2}{2}+\text{c}$ $\Rightarrow\ \ \text{y}\sin\text{x}=2\text{x}^2+\text{c}\ \ ...{(\text{i})}$
$\text{Now Putting y}=0,\text{x}=\frac{\pi}{2}\ \text{in eq. (i)},$ $0=2.\frac{\pi^{2}}{4}+\text{c}\ \ \Rightarrow\ \ \text{c}=\frac{-\pi^{2}}{2}$
$\text{Putting c}=\frac{-\pi^2}{2}\ \text{in eq. (i)},\ \ \text{y}\sin\text{x}=2\text{x}^2-\frac{\pi^2}{2}$

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