Solve each of the following initial value problems:(i) y'+y=ex, y0=12(ii) xdydx-y=log x, y1=0(iii) dydx+2y=e-2x sin x, y0=0(iv) xdydx-y=x+1e-x, y1=0(v) 1+y2 dx+x-e-tan-1y dx=0, y0=0(vi) dydx+y tan x=2x+x2 tan x, y0=1(vii) xdydx+y=x cos x+sin x, yπ2=1(viii) dydx+y cot x=4x cosec x, yπ2=0(ix) dydx+2y tan x=sin x; y=0 when x=π3(x) dydx-3y cot x=sin 2x; y=2 when x=π2(xi)(xii)

Solve each of the following initial value problems:(i) y'+y=ex, y0=12(

1.	Solve each of the following initial value problems:(i) y'+y=ex, y0=12(ii) xdydx-y=log x, y1=0(iii) dydx+2y=e-2x sin x, y0=0(iv) xdydx-y=x+1e-x, y1=0(v) 1+y2 dx+x-e-tan-1y dx=0, y0=0(vi) dydx+y tan x=2x+x2 tan x, y0=1(vii) xdydx+y=x cos x+sin x, yπ2=1(viii) dydx+y cot x=4x cosec x, yπ2=0(ix) dydx+2y tan x=sin x; y=0 when x=π3(x) dydx-3y cot x=sin 2x; y=2 when x=π2(xi)(xii)
Answer» Solve each of the following initial value problems: (i) $y' + y = e^{x}, y (0) = \frac{1}{2}$ (ii) $x \frac{d y}{d x} - y = \log x, y (1) = 0$ (iii) $\frac{d y}{d x} + 2 y = e^{- 2 x} \sin x, y (0) = 0$ (iv) $x \frac{d y}{d x} - y = (x + 1) e^{- x}, y (1) = 0$ (v) $(1 + y^{2}) d x + (x - e^{- \tan^{- 1} y}) d x = 0, y (0) = 0$ (vi) $\frac{d y}{d x} + y \tan x = 2 x + x^{2} \tan x, y (0) = 1$ (vii) $x \frac{d y}{d x} + y = x \cos x + \sin x, y (\frac{π}{2}) = 1$ (viii) $\frac{d y}{d x} + y \cot x = 4 x cosec x, y (\frac{π}{2}) = 0$ (ix) $\frac{d y}{d x} + 2 y \tan x = \sin x; y = 0 when x = \frac{π}{3}$ (x) $\frac{d y}{d x} - 3 y \cot x = \sin 2 x; y = 2 when x = \frac{π}{2}$ (xi) (xii)