Question

TOPIC: DIFFERENTIAL EQUATIONS

Verify that the function is general solution of nonhomogenous differential equation.

Function is \(\sf y=c_1•cos(x) + c_2•sin(x)+ x•sin(x) + (cos(x))ln(cos(x))\)
DE: \(\sf y"+y=sec\ x\)
Interval: \(\sf (-\frac{\pi}{2}, \frac{\pi}{2})\)

Answer 1

from the theory, the general solution is the sum of the solution of the homogenous differential equation \(\sf y_c\) and a particular solution of the nonhomogenous differential equation \(\sf y_p\): \(\sf y=y_c + y_p\)

So I am expecting to get:
\(\sf y_c=c_1•cos(x) + c_2•sin(x)\)
\(\sf y_p=x•sin(x) + (cos(x))ln(cos(x))\)


So I solved for \(\sf y_c\) first:
\(\sf y"+y=0\\ let\ y=e^{mx} \\ ... \\ m^2+m= 0\\ m(m+1)=0\\m=-1,\ m=0 \\ hence, y_c=c_1e^{-x} + c_2\)

which doesn't match with what I am expecting.....

(P.S. We haven't covered the Variation of parameters yet, so I can't use that)

Answer 2

\(m^2 \color{red}{+ 1}\)

Answer 3

ohhhh right , thanks.