Solve the DE Solve the differential equation by variation of

Solve the DE.

Solve the differential equation by variation of parameters. y\" + y = tan x

Solution

First we solve associated homogeneous ode

y\'\'+y=0

General solution is

y=A sin(x)+B cos(x)

Based on this the guess for particular oslution is

yp=P(x) sin(x)+Q(x)cos(x)

with the constraint: P\' sin(x)+Q\' cos(x)=0

so, Q\'=-P\' tan(x)

yp\'=P cos(x)-Q sin(x)

yp\'\'=-yp+P\' cos(x)-Q\' sin(x)

Substituting gives

P\' cos(x)-Q\' sin(x)=tan(x)

Q\'=-P\' tan(x)

P\' cos(x)+P\' sin^2(x)/cos(x)=tan(x)

P\'=sin(x)

P=-cos(x)

Q\'=-P\' tan(x)=-sin^2(x)/cos(x)=(cos^2(x)-1)/cos(x)=cos(x)-sec(x)

Q=sin(x)-log(sec(x)+tan(x))