Question

For each of the differential equations given, find a particular solution satisfying the given condition.
1. $\frac{dy}{dx}+2y\tan x=\sin x:y=0$ when $x=\frac{\pi }{3}$
2. $(1+x^2)\frac{dy}{dx}+2xy=\frac{1}{1+x^2};y=0$ when $x=1$
3. $\frac{dy}{dx}-3y\cot x=\sin 2x;y=2$ when $x=\frac{\pi }{2}$

Answer 1

1) $\frac{dy}{dx}+2y\tan x=\sin x;y=0$, when $x=\frac{\pi }{3}$

$4\left(x\right)={\int }_{e}2\tan x.dx=2ln\left(x\right)={\sec }^{2}x$

${\sec }^{2}x(y^1+2\tan xy)=\sin x.se{c}^{2}x$

$({\sec }^{2}x{)}^1=\sin x.{\sec }^{2}x$

$\sec 62x.y=\int x.{\sec }^{2}x.dx$

$\int \sin x.\frac{1}{{\cos }^{2}x}.dx$

$\cos x=+$

$\sin x.dx=-d+$

$=\int \frac{-dt}{+2}=\frac{1}{+}=\frac{1}{\cos x}+c$

$y=\cos x+c{\cot }^{2}x$

$y(x=\frac{\pi }{3})=0$

$o=\cos \left(\frac{\pi }{3}\right)+c{\cos }^2\left(\frac{\pi }{3}\right)$

$o=\frac{1}{2}+c.\frac{1}{4}$

$c=2$

$y\left(x\right)=\cos \left(x\right)+{\cos }^{2}x$

2) $(1+x^2)\frac{dy}{dx}+2xy=\frac{1}{1+x^2};y(x=1)=0$

$\frac{dy}{dx}=\frac{2x}{1+x62}y=\frac{1}{(1+x^2{)}^2}$

$4\left(x\right)={\int }_e\frac{2x}{1+x^2}.dx$

$={\int }_e\frac{dt}{1+1}=eln(1+1)$

$=(1+x^2)$

$(1+x^2)(y^1+\frac{2x}{1+x^2}y)=\frac{1}{(1+x^2{)}^2}.(1+x^2)$

$\left(\right(1+x^2)y{)}^1=\frac{1}{1+x^2}$

$(1+x^2)y=\int \frac{1}{1+x^2}.dx$

$(1+x^2)y=\tan \left(x\right)+c$

$y=\frac{\tan \left(x\right)}{1+n^2}+\frac{c}{1+x^2}$

$y(n=1)=o\Rightarrow 0=\frac{\frac{\pi }{2}}{2}+\frac{c}{2}$

$c=-\frac{\pi }{4}$

$y=\frac{\tan \left(x\right)}{1+x^2}-\frac{\pi }{4(1+x^2)}$

3) $\frac{dy}{dx}-3y\cot x=\sin x;\left(y\right(x=\frac{x}{2})=2$

$4\left(x\right)={\int }_e-3\cot x.dx=-3\log |\sin x|=\frac{1}{{\sin }^{3}x}$

$\frac{1}{{\sin }^{2}x}(y^2-3y\cot x)=\frac{\sin \$2x}{{\sin }^{3}x}$

${\left[\frac{y}{{\sin }^{3}z}\right]}^1=\frac{\sin x^{2}x}{{\sin }^{3}x}$

$\frac{y}{{\sin }^{3}x}=\int \frac{2\cos x}{\sin x}.dx$

$\sin x=+\Rightarrow \cos x.dx=d+$

$\int \frac{2dt}{+2}$

$\frac{-2}{+}=\frac{-2}{\sin x}=c$

$y=-2{\sin }^{2}x+c{\sin }^{3}x$

$y(x=\frac{\pi }{2})=2$

$2=-2+c$

$c=4$

$y\left(x\right)=-2{\sin }^{2}x+4{\sin }^{3}x$