Find the general solution for each of the following differential equations. $\frac{d y}{d x}+2 y \cot x=3 x^{2} \operatorname{cosec}^{2} x$

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CBSE | Class 12 | Linear Differential Equations | Maths | RS Aggarwal

Find the general solution for each of the following differential equations. $\frac{d y}{d x}+2 y \cot x=3 x^{2} \operatorname{cosec}^{2} x$

Solution:

$\frac{d y}{d x}+2 \mathrm{y}(\cot x)=3 x^{2} \operatorname{cosec}^{2} x\dots (1)$
General solution for the differential equation in the form of
$\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$
General solution is given by,
$y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c$
Where, integrating factor,
$\text { I. } F .=e \int^{p d x}$
Comparing equation (1) with
$\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$
Where, $P=2 \operatorname{Cotx}$ and $Q=3 x^{2} \operatorname{cosec}^{2} x$
Therefore, integrating factor is
$I . F .=e \int^{p d x}=\sin ^{2} \mathrm{x}$
General solution is
$\begin{array}{l} y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c \\ \therefore y \cdot\left(\sin ^{2} x\right)=\int\left(3 x^{2} \operatorname{cosec}^{2} x\right) \cdot\left(\sin ^{2} x\right) \mathrm{d} x+c \\ \therefore y \cdot\left(\sin ^{2} x\right)=3 \frac{x^{3}}{3}+c \\ \therefore y \cdot\left(\sin ^{2} x\right)=x^{3}+c \\ \end{array}$