ODE/(D^4 - 1) y = sin x
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Theorem
The second order ODE:
- $(1): \quad \paren {D^4 - 1} y' = \sin x$
has a general solution:
- $y = C_1 e^x + C_2 e^{-x} + C_3 \sin x + C_4 \cos x + \dfrac {x \cos x} 4$
Proof
First we solve the reduced equation of $(1)$:
- $(2): \quad \paren {D^4 - 1} y' = 0$
The auxiliary equation of $(1)$ is:
- $(3): \quad: m^4 - 1 = 0$
From Complex $4$th Roots of Unity, the roots of $(2)$ are:
- $m_1 = 1$
- $m_2 = i$
- $m_3 = -1$
- $m_4 = -i$
So from Solution of Constant Coefficient Linear nth Order ODE, the general solution of $(2)$ is:
- $y_g = C_1 e^x + C_2 e^{-x} + C_3 \sin x + C_4 \cos x$
Because $\sin x$ is already a solution of $(2)$, we try:
- $y = A x e^{i x}$
with a view to taking the real part in due course.
\(\ds y\) | \(=\) | \(\ds A x e^{i x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y'\) | \(=\) | \(\ds A e^{i x} + A i x e^{i x}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds A \paren {1 + i x} e^{i x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds A \paren {1 + i x} i e^{i x} + A i e^{i x}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds A \paren {2 i - x} e^{i x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y^{\paren 3}\) | \(=\) | \(\ds A \paren {2 i - x} i e^{i x} - A e^{i x}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds A \paren {-3 - i x} e^{i x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y^{\paren 4}\) | \(=\) | \(\ds A \paren {-3 - i x} i e^{i x} - i e^{i x}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds A \paren {x - 4 i} e^{i x}\) |
It follows that
- $-4 i A - 1$
and so:
- $A = -\dfrac 1 4$
Thus a particular solution to $(1)$ can be given as:
\(\ds y_p\) | \(=\) | \(\ds \map \Im {\dfrac 1 4 i x e^{i x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x \cos x} 4\) |
and the general solution is:
- $y = C_1 e^x + C_2 e^{-x} + C_3 \sin x + C_4 \cos x + \dfrac {x \cos x} 4$
$\blacksquare$
Sources
- 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 3$. Equations of higher order and systems of first order equations: $\S 3.1$ The $n$th order equation: Example $1$