Linear Second Order ODE/y'' + y = cosecant x
Theorem
The second order ODE:
- $(1): \quad y + y = \csc x$
has the general solution:
- $y = C_1 \sin x + C_2 \cos x - x \cos x + \sin x \map \ln {\sin x}$
Proof
It can be seen that $\paren 1$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
- $y + p y' + q y = \map R x$
where:
- $p = 0$
- $q = 1$
- $\map R x = \csc x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
- $\paren 2: \quad y + y = 0$
From Linear Second Order ODE: $y + y = 0$, this has the general solution:
- $y_g = C_1 \sin x + C_2 \cos x$
It remains to find a particular solution $y_p$ to $\paren 1$.
Expressing $y_g$ in the form:
- $y_g = C_1 \map {y_1} x + C_2 \map {y_2} x$
we have:
\(\ds \map {y_1} x\) | \(=\) | \(\ds \sin x\) | ||||||||||||
\(\ds \map {y_2} x\) | \(=\) | \(\ds \cos x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {y_1'} x\) | \(=\) | \(\ds \cos x\) | Derivative of Sine Function | ||||||||||
\(\ds \map {y_2'} x\) | \(=\) | \(\ds -\sin x\) | Derivative of Cosine Function |
By the Method of Variation of Parameters, we have that:
- $y_p = v_1 y_1 + v_2 y_2$
where:
\(\ds v_1\) | \(=\) | \(\ds -\int \frac {y_2 \map R x} {\map W {y_1, y_2} } \rd x\) | ||||||||||||
\(\ds v_2\) | \(=\) | \(\ds \int \frac {y_1 \map R x} {\map W {y_1, y_2} } \rd x\) |
where $\map W {y_1, y_2}$ is the Wronskian of $y_1$ and $y_2$.
We have that:
\(\ds \map W {y_1, y_2}\) | \(=\) | \(\ds y_1 {y_2}' - y_2 {y_1}'\) | Definition of Wronskian | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin x \paren {- \sin x} - \cos x \cos x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\paren {\sin^2 x + \cos^2 x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -1\) |
Hence:
\(\ds v_1\) | \(=\) | \(\ds \int -\frac {y_2 \map R x} {\map W {y_1, y_2} } \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\int \frac {\cos x \csc x} {-1} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cos x} {\sin x} \rd x\) | Definition of Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {\sin x}\) | Primitive of $\cot x$ |
\(\ds v_2\) | \(=\) | \(\ds \int \frac {y_1 \map R x} {\map W {y_1, y_2} } \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\sin x \csc x} {-1} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\int \frac {\sin x} {\sin x} \rd x\) | Definition of Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds -x\) |
It follows that:
- $y_p = \sin x \map \ln {\sin x} - x \cos x$
So from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution:
- $y = y_g + y_p = C_1 \sin x + C_2 \cos x + \sin x \map \ln {\sin x} - x \cos x$
is the general solution to $\paren 1$.
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.19$: The Method of Variation of Parameters: Example $1$