Linear Second Order ODE/y'' - 5 y' + 6 y = cos x + sin x

Theorem

The second order ODE:

$(1): \quad y - 5 y' + 6 y = \cos x + \sin x$

has the general solution:

$y = C_1 e^{2 x} + C_2 e^{3 x} + \dfrac {\cos x} 5$

Proof

It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:

$y + p y' + q y = \map R x$

where:

$p = -5$

$q = 5$

$\map R x = \cos x + \sin x$

First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:

$y - 5 y' + 6 y = 0$

From Linear Second Order ODE: $y - 5 y' + 6 y = 0$, this has the general solution:

$y_g = C_1 e^{2 x} + C_2 e^{3 x}$

It remains to find a particular solution $y_p$ to $(1)$.

We have that:

$\map R x = e^{-x} \cos x$

From the Method of Undetermined Coefficients for Sine and Cosine:

$y_p = A \cos x + B \sin x$

where $A$ and $B$ are to be determined.

Hence:

\(\ds y_p\)

\(=\)

\(\ds A \cos x + B \sin x\)

\(\ds \leadsto \ \ \)

\(\ds {y_p}'\)

\(=\)

\(\ds -A \sin x + B \cos x\)

Derivative of Sine Function, Derivative of Cosine Function

\(\ds \leadsto \ \ \)

\(\ds {y_p}\)

\(=\)

\(\ds -A \cos x - B \sin x\)

Derivative of Sine Function, Derivative of Cosine Function

Substituting into $(1)$:

\(\ds \paren {-A \cos x - B \sin x} - 5 \paren {-A \sin x + B \cos x} + 6 \paren {A \cos x + B \sin x}\)

\(=\)

\(\ds \cos x + \sin x\)

\(\ds \leadsto \ \ \)

\(\ds \paren {5 A - 5 B} \cos x\)

\(=\)

\(\ds \cos x\)

equating coefficients

\(\ds \paren {5 B + 5 A} \sin x\)

\(=\)

\(\ds \sin x\)

\(\ds \leadsto \ \ \)

\(\ds 10 A\)

\(=\)

\(\ds 2\)

\(\ds 10 B\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds A\)

\(=\)

\(\ds \dfrac 1 5\)

\(\ds B\)

\(=\)

\(\ds 0\)

Hence the result:

$y_p = \dfrac {\cos x} 5$

So from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution:

$y = y_g + y_p = C_1 e^{2 x} + C_2 e^{3 x} + \dfrac {\cos x} 5$

is the general solution to $(1)$.

$\blacksquare$

Sources

• 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: Problems for Chapter $1$: $7$