If (dy/dx)+y cos x=(1/2) sin 2x, y(0)=1, then y((π/2))=

Question

If (dy/dx)+y cos x=(1/2) sin 2x, y(0)=1, then y((π/2))= (A) 1 (B) e (C) (1/e) (D) (2/e)

Tardigrade Admin · Accepted Answer

It is linear differential equation with I.F. = textexp ∫ cos x dx=esin x. The solution is, yesinx=∫ sin x cos x esin x dx ⇒ yesinx=(sinx-1)esinx+c ⇒ y=sin x-1+ce-sin x y(0)=1 ⇒ c=2 ⇒ y=sin x-1+2e-sin x ∴ y((π/2))=(2/e).

Q. If $\frac{d y}{d x} + y cos x = \frac{1}{2} s in 2 x$ , $y (0) = 1$ , then $y (\frac{π}{2}) =$

$1$

$e$

$\frac{1}{e}$

$\frac{2}{e}$

Q. If dxdy​+ycosx=21​sin2x, y(0)=1, then y(2π​)=

1

e

e1​

e2​

It is linear differential equation with I.F. =exp∫cosxdx=esinx. The solution is, yesinx=∫sinxcosxesinxdx ⇒yesinx=(sinx−1)esinx+c ⇒y=sinx−1+ce−sinx y(0)=1 ⇒c=2 ⇒y=sinx−1+2e−sinx ∴y(2π​)=e2​.

Q. If $\frac{d y}{d x} + y cos x = \frac{1}{2} s in 2 x$ , $y (0) = 1$ , then $y (\frac{π}{2}) =$

$1$

$e$

$\frac{1}{e}$

$\frac{2}{e}$