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

Theorem

$(1): \quad y' ' + \map P x y' + \map Q x y = \map {R_1} x$

$(2): \quad y' ' + \map P x y' + \map Q x y = \map {R_2} x$

Then $\map y x = \map {y_1} x + \map {y_2} x$ is a particular solution of the linear second order ODE:

$(3): \quad y' ' + \map P x y' + \map Q x y = \map {R_1} x + \map {R_2} x$

$\text {(4)}: \quad$

$\ds {y_1}' ' + \map P x {y_1}' + \map Q x y_1$

$=$

$\ds \map {R_1} x$

as $y_1$ is a particular solution to $(1)$

$\text {(5)}: \quad$

$\ds {y_2}' ' + \map P x {y_2}' + \map Q x y_2$

$=$

$\ds \map {R_2} x$

as $y_2$ is a particular solution to $(2)$

$\ds \leadsto \ \ $

$\ds \paren { {y_1}' ' + {y_2}' '} + \map P x \paren { {y_1}' + {y_2}'} + \map Q x \paren {y_1 + y_2}$

$=$

$\ds \map {R_1} x + \map {R_2} x$

adding $(4)$ and $(5)$

$\ds \leadsto \ \ $

$\ds \paren {y_1 + y_2}' ' + \map P x \paren {y_1 + y_2}' + \map Q x \paren {y_1 + y_2}$

$=$

$\ds \map {R_1} x + \map {R_2} x$

Hence the result.

$\blacksquare$