Method of Undetermined Coefficients/Sine and Cosine/Particular Solution/i b is Root of Auxiliary Equation
Proof Technique
Consider the nonhomogeneous linear second order ODE with constant coefficients:
- $(1): \quad y + b^2 y = \alpha \sin b x + \beta \cos b x$
The Method of Undetermined Coefficients can be used to find a particular solution to $(1)$ in the following manner.
Method and Proof
Let $\map {y_g} x$ be the general solution to:
- $y + b^2 y = 0$
From General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution:
- $\map {y_g} x + \map {y_p} x$
is the general solution to $(1)$.
It remains to find $\map {y_p} x$.
Consider the auxiliary equation to $(1)$:
- $(2): \quad m^2 + b^2 = 0$
We have that $i b$ is a root of $(2)$.
Trigonometric Form
Assume that there is a particular solution to $(1)$ of the form:
- $y_p = x \paren {A \sin b x + B \cos b x}$
We have:
\(\ds \frac {\d} {\d x} y_p\) | \(=\) | \(\ds x \paren {b A \cos b x - b B \sin b x} + \paren {A \sin b x + B \cos b x}\) | Derivative of Sine Function, Derivative of Cosine Function, Product Rule for Derivatives | |||||||||||
\(\ds \frac {\d^2} {\d x^2} y_p\) | \(=\) | \(\ds x \paren {-b^2 A \sin b x - b^2 B \cos b x} + \paren {b A \cos b x - b B \sin b x} + \paren {b A \cos b x - b B \sin b x}\) | Derivative of Sine Function, Derivative of Cosine Function, Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds x \paren {-b^2 A \sin b x - b^2 B \cos b x} + 2 \paren {b A \cos b x - b B \sin b x}\) |
Inserting into $(1)$:
\(\ds x \paren {-b^2 A \sin b x - b^2 B \cos b x} + 2 \paren {b A \cos b x - b B \sin b x} + b^2 x \paren {A \sin b x + B \cos b x}\) | \(=\) | \(\ds \alpha \sin b x + \beta \cos b x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \paren {b A \cos b x - b B \sin b x}\) | \(=\) | \(\ds \alpha \sin b x + \beta \cos b x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 b A \cos b x\) | \(=\) | \(\ds \beta \cos b x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -2 b B \sin b x\) | \(=\) | \(\ds \alpha \sin b x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 b A\) | \(=\) | \(\ds \beta\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -2 b B\) | \(=\) | \(\ds \alpha\) |
Hence $A$ and $B$ can be expressed in terms of $\alpha$ and $\beta$:
\(\ds \leadsto \ \ \) | \(\ds A\) | \(=\) | \(\ds \frac \beta {2 b}\) | |||||||||||
\(\ds B\) | \(=\) | \(\ds -\frac \alpha {2 b}\) |
Hence:
- $y_p = \dfrac {\beta x \sin b x} {2 b} - \dfrac {\alpha x \cos b x} {2 b}$
Exponential Form
Assume that there is a particular solution to $(1)$ of the form:
- $y_p = x \paren {A \sin b x + B \cos b x}$
From Euler's Formula:
- $\cos b x + i \sin b x = e^{i b x}$
and so:
- $x \paren {A \sin b x + B \cos b x}$ is the real part of $x \paren {A - i B} \paren {\cos b x + i \sin b x} = x \paren {A - i B} e^{i b x}$
It is assumed that $A$, $B$, $p$ and $q$ are all real numbers.
Suppose we have found a solution $y$ of $(1)$ where:
- $\map f x = \map {f_1} x + i \, \map {f_2} x$
where $\map y x$ and $\map f x$ are complex-valued.
Letting $\map y x = \map {y_1} x + \map {y_2} x$, where $y_1$ and $y_2$ are the real and imaginary parts of $\map y x$, we have:
- ${y_1} + p {y_1}' + q y_1 + i \paren { {y_2} + p {y_2}' + q y_2} = \map {f_1} x + i \, \map {f_2} x$
Equating real parts:
- ${y_1} + p {y_1}' + q y_1 = \map {f_1} x$
Equating imaginary parts:
- ${y_2} + p {y_2}' + q y_2 = \map {f_2} x$
Thus if $y$ is a particular solution to $(1)$ when the right hand side is $\map f x$:
- $\map \Re y$ is a particular solution to $(1)$ when the right hand side is $\map \Re {\map f x}$
- $\map \Im y$ is a particular solution to $(1)$ when the right hand side is $\map \Im {\map f x}$
So to find a particular solution when the right hand side is $K \cos x$ or $K \sin x$, we can first find a particular solution when the right hand side is $K e^{i b x}$ and then take its real part or imaginary part as necessary.
Hence, when we have $A \cos b x + B \sin b x$ on the right hand side:
- replace it with $x \paren {A - i B} e^{i b x}$
- use the Method of Undetermined Coefficients for Exponential functions
and then take its real part.
$\blacksquare$
Sources
- 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 2$. The second order equation: $\S 2.5$ Particular solution: trigonometric $\map f x$