Find a particular solution satisfying the given condition for each of the following differential equations. $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \cot \mathrm{x}=4 \mathrm{x} \operatorname{cosec} \mathrm{x}$, given that $y=0$ when $x=\frac{\pi}{2}$

Home » Find a particular solution satisfying the given condition for each of the following differential equations. , given that when

Find a particular solution satisfying the given condition for each of the following differential equations. $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \cot \mathrm{x}=4 \mathrm{x} \operatorname{cosec} \mathrm{x}$ , given that $y=0$ when $x=\frac{\pi}{2}$

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

Find a particular solution satisfying the given condition for each of the following differential equations. $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \cot \mathrm{x}=4 \mathrm{x} \operatorname{cosec} \mathrm{x}$ , given that $y=0$ when $x=\frac{\pi}{2}$

Solution:

$\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \cdot \operatorname{Cot} \mathrm{x}=4 \mathrm{x} \operatorname{Cosec} \mathrm{x}^{--}-\mathrm{c}-\mathrm{c}-\mathrm{c}(1)\dots (1)$
General solution for the differential equation in the form of $\frac{d y}{d x}+\mathrm{Py}=$ Qis given byy. $(I . F .)=\int Q \cdot(I . F .) d x+c$ Where, integrating factor,
$\text { I. } F .=e \int^{p d x}$
Equation (1) is of the form
$\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$
Where, $\mathrm{P}=\operatorname{Cotx}$ and $\mathrm{Q}=4 \mathrm{x}$ Cosecx
Therefore, integrating factor is
$\text { I. F. }=e \int^{p d x}=e \int \cot x d x=e^{\log \sin x}=\sin \mathrm{x}$
General solution is
$\begin{array}{l} y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c \\ y \cdot \operatorname{Sin} x=\int 4 x \operatorname{Cosec} x \operatorname{Sin} x d x+c \\ y \cdot \operatorname{Sin} x=4 \int_{x} \cdot d x+c \\ y \cdot \sin x=4 \frac{x^{2}}{2}+c \end{array}$
$y \cdot \operatorname{Sin} x=2 x^{2}+c$ (general equation)
For particular solution put $\mathrm{y}=0$ and $\mathrm{x}=\pi / 2$ in above equation, $c=-\pi^{2} / 2$
Therefore, particular solution is
$\mathrm{y} \cdot \operatorname{Sin} \mathrm{x}=2 \mathrm{x}^{2}-\frac{\pi^{2}}{2}$