(x+y) (x-y) =0

Solve each of the following initial value problems:
(i) $y\text{'}+y=e^x,y\left(0\right)=\frac{1}{2}$

(ii) $x\frac{dy}{dx}-y=\log x,y\left(1\right)=0$

(iii) $\frac{dy}{dx}+2y=e^{-2x}\sin x,y\left(0\right)=0$

(iv) $x\frac{dy}{dx}-y=\left(x+1\right)e^{-x},y\left(1\right)=0$

(v) $\left(1+y^2\right)dx+\left(x-e^{-{\tan }^{-1}y}\right)dx=0,y\left(0\right)=0$

(vi) $\frac{dy}{dx}+y\tan x=2x+x^2\tan x,y\left(0\right)=1$

(vii) $x\frac{dy}{dx}+y=x\cos x+\sin x,y\left(\frac{\mathrm{\pi }}{2}\right)=1$

(viii) $\frac{dy}{dx}+y\cot x=4x\mathrm{cosec}x,y\left(\frac{\mathrm{\pi }}{2}\right)=0$

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

(x) $\frac{dy}{dx}-3y\cot x=\sin 2x;y=2\mathrm{when}x=\frac{\mathrm{\pi }}{2}$
(xi) $\frac{dy}{dx}+y\cot x=2\cos x,y\left(\frac{\mathrm{\pi }}{2}\right)=0$
(xii) $dy=\cos x\left(2-y\cos ecx\right)dx$
(xiii) $\tan x\left(\frac{dy}{dx}\right)=2x\tan x+x^2-y;\tan x\ne 0$given that y = 0 when $x=\frac{\mathrm{\pi }}{2}$.

Solve each of the following initial value problems:
(i) (x² + y²) dx = 2xy dy, y (1) = 0

(ii) $x{e}^{y/x}-y+x\frac{dy}{dx}=0,y\left(e\right)=0$

(iii) $\frac{dy}{dx}-\frac{y}{x}+\mathrm{cosec}\frac{y}{x}=0,y\left(1\right)=0$

(iv) (xy − y²) dx − x² dy = 0, y(1) = 1

(v) $\frac{dy}{dx}=\frac{y\left(x+2y\right)}{x\left(2x+y\right)},y\left(1\right)=2$

(vi) (y⁴ − 2x³ y) dx + (x⁴ − 2xy³) dy = 0, y (1) = 1

(vii) x (x² + 3y²) dx + y (y² + 3x²) dy = 0, y (1) = 1

(viii) $\left\{x{\sin }^2\left(\frac{y}{x}\right)-y\right\}dx+xdy=0,y\left(1\right)=\frac{\mathrm{\pi }}{4}$

(ix) $x\frac{dy}{dx}-y+x\sin \left(\frac{y}{x}\right)=0,y\left(2\right)=x$

Solve each of the following initial value problems:
(i) $y\text{'}+y=e^x,y\left(0\right)=\frac{1}{2}$

(ii) $x\frac{dy}{dx}-y=\log x,y\left(1\right)=0$

(iii) $\frac{dy}{dx}+2y=e^{-2x}\sin x,y\left(0\right)=0$

(iv) $x\frac{dy}{dx}-y=\left(x+1\right)e^{-x},y\left(1\right)=0$

(v) $\left(1+y^2\right)dx+\left(x-e^{-{\tan }^{-1}y}\right)dx=0,y\left(0\right)=0$

(vi) $\frac{dy}{dx}+y\tan x=2x+x^2\tan x,y\left(0\right)=1$

(vii) $x\frac{dy}{dx}+y=x\cos x+\sin x,y\left(\frac{\mathrm{\pi }}{2}\right)=1$

(viii) $\frac{dy}{dx}+y\cot x=4x\mathrm{cosec}x,y\left(\frac{\mathrm{\pi }}{2}\right)=0$

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

(x) $\frac{dy}{dx}-3y\cot x=\sin 2x;y=2\mathrm{when}x=\frac{\mathrm{\pi }}{2}$

Solve each of the following initial value problems:
(i) $y\text{'}+y=e^x,y\left(0\right)=\frac{1}{2}$

(ii) $x\frac{dy}{dx}-y=\log x,y\left(1\right)=0$

(iii) $\frac{dy}{dx}+2y=e^{-2x}\sin x,y\left(0\right)=0$

(iv) $x\frac{dy}{dx}-y=\left(x+1\right)e^{-x},y\left(1\right)=0$

(v) $\left(1+y^2\right)dx+\left(x-e^{-{\tan }^{-1}y}\right)dx=0,y\left(0\right)=0$

(vi) $\frac{dy}{dx}+y\tan x=2x+x^2\tan x,y\left(0\right)=1$

(vii) $x\frac{dy}{dx}+y=x\cos x+\sin x,y\left(\frac{\mathrm{\pi }}{2}\right)=1$

(viii) $\frac{dy}{dx}+y\cot x=4x\mathrm{cosec}x,y\left(\frac{\mathrm{\pi }}{2}\right)=0$

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

(x) $\frac{dy}{dx}-3y\cot x=\sin 2x;y=2\mathrm{when}x=\frac{\mathrm{\pi }}{2}$
(xi)
(xii)