Combination of Solutions to Non-Homogeneous LSOODE with same Homogeneous Part

Theorem
Let $y_1 \left({x}\right)$ be a particular solution of the linear second order ODE:
 * $(1): \quad y'' + P \left({x}\right) y' + Q \left({x}\right) y = R_1 \left({x}\right)$

Let $y_2 \left({x}\right)$ be a particular solution of the linear second order ODE:
 * $(2): \quad y'' + P \left({x}\right) y' + Q \left({x}\right) y = R_2 \left({x}\right)$

Then $y \left({x}\right) = y_1 \left({x}\right) + y_2 \left({x}\right)$ is a particular solution of the linear second order ODE:
 * $(3): \quad y'' + P \left({x}\right) y' + Q \left({x}\right) y = R_1 \left({x}\right) + R_2 \left({x}\right)$

Proof
Hence the result.