Answer to Question #134894 in Differential Equations for Nikhil

Question

Use the method of reduction of order to solve the following equation
(D^2+1)y=cosecx

Answer 1

These equation cannot be solved by method of reduction of order, for using this method we must have at least one solution of the given equation.

These can be solved by using Method of variation of parameters,

(D$^2$ +1)y = cosec x

The auxiliary equation is $(m^2+1)=0$

$m=\pm i$

Therefore complimentary function CF=$c_1cos x+c_2sinx$

i.e. =$c_1f_1(x)+c_2f_2(x)$

where

$f_1(x)=cos x$

$f_2(x)=sinx$

$f'_1(x)=-sinx$

$f'_2(x)=cosx$

R=cosec x

$f_1f'_2-f'_1f_2=cos^2x+sin^2x=1$

P=$-\int \frac{f_2R}{(f_1f'_2-f'_1f_2)}dx$

=$-\int sinx cosecx$ dx

$=-\int dx$

=-x

similarly

Q=$\int \frac{f_1R}{(f_1f'_2-f'_1f_2)}dx$

=$\int cos x cosecxdx$

=$\int \frac{cosx}{sinx}dx$

$=\int cotxdx$

=log(sin x)

P.I.=$Pf_1+Qf_2$

=$-xcosx+sinxlog(sinx)$

complete solution

y=c.f.+p.I.

=$c_1cosx+c_2sinx-xcosx+sinxlog(sinx)$