let y_p be a particular solution of the nonhomogeneous eqn y'' + py' + qy = f(x) and let y_c be a solution of its associated homogeneous eqn. show that y=y_c + y_p is a solution of the given nonhomogeneous eqn.
let y_p be a particular solution of the nonhomogeneous eqn y'' + py' + qy = f(x) and let y_c be a solution of its associated homogeneous eqn. show that y=y_c + y_p is a solution of the given nonhomogeneous eqn.
On the basis of your hypotheses is...
$\displaystyle y_{c}^{''} + p \cdot y_{c}^{'} + q\cdot y_{c} =0$
$\displaystyle y_{p}^{''} + p \cdot y_{p}^{'} + q\cdot y_{p} = f(x)$
... so that setting $\displaystyle y(x)= y_{c} (x) + y_{p} (x)$ , because the derivative is a linear operator, you have...
$\displaystyle y^{''} + p \cdot y^{'} + q\cdot y = f(x)$
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$