Question

Solve by variation of parameters` ((d^2y)/dx^2+1)y=1/(1+sin x)`

Answer 1

put `d(d/dx)=D`

`(D^2+1)y=1/(1+sin x)`

For complementary solution, 

`f(D)=0` 

∴ ` (D^2+1)=0` 

Roots are : D = 𝒊 ,−𝒊
Roots of given diff. eqn are complex.
The complementary solution of given diff. eqn is given by, 

∴ `y_c=c_1 cos x+c_2 sin x`

For particular solution ,
By method of variation of parameters, 

`y_p=y_1p_1+y_2p_2 `    Where` p_1( int-y_2 x)/w  dx` 

                                         `  p_2= int (y_1 x)/w` 

                                        w= `|[ y_1,y_2],[y'_1,y'_2]|` 

w=`|[cos x,sin x],[-sin x, cos x]| = cos^2x+sin^2 x=1`

`p_1= int ^(-y_2 x)/w dx= int - sin x/1. 1/1+sin x dx= - int (sin x (1-sin x))/(1+sin x (1-sin x)) dx`       

`y_p=- [sec x-tan x+x]cos x +log(1+sin x )sin x`   

The general solution of given diff. eqn is given by , 

`y_g=y_c+y_p=c_1cos x+c-2 sin x-[sec x-tan x+x] cosx+log (1+sin x)sin x`