Question

solve the undetermined coefficient y''+y=x

Answer 1

given differential equation,

y"+y=x

homogeneous equation,

y"+y=0

solution is given by,

y=acosx+bsinx

where a and b and constants.

The undetermined coefficient function is 1 and x.

$y_p=e+fx\newline y'_p=f\newline y''_p=0$

where e and f and constants.

substitute the above values in the given differential equation,

$e+fx=x$

comparing both side ,we get,

e=0, f=1

therefore ,general solution is y=a cosx +b sinx+x