2.4: Superposition and General Solutions

Reading: Notes on Diffy Q’s Section 2.1 and Section 2.5

A Recap of Solution Methods

Finding a Homogeneous Solution

We have developed methods for finding the general homogeneous solution, $y_h(t)$, to a second order differential equation of the form

\[ ay'' + b y' + cy =0 , \qquad \mbox{for constants $a$, $b$, and $c$:}\]

Write the differential equation in standard form, $ay'' + b y' + cy =0$
Find the roots of the characteristic equation $ar^2+br+c=0$.

If there are two distinct real roots $r_1$ and $r_2$, the general solution is $y_h(t) = C_1e^{r_1t} + C_2e^{r_2t}$.
If there is one repeated real root $r_1$, the general solution is $y_h(t) = C_1e^{r_1t} + C_2te^{r_1t}$.
If there are two complex-cojugate solutions $r=\alpha \pm i \beta$, the general solution is $y_h(t) = C_1e^{\alpha t} \cos{(\beta t)} + C_2e^{\alpha t} \sin{(\beta t)}$.

Finding a Particular Solution

We have also discussed how to find the particular solution, $y_p(t)$ to a nonhomogeneous second order differential equation of the form

\[ay'' + b y' + cy =f(t).\]

Based on the form of $f(t)$, we guess $y_p(t)$ will have a similar form.

Table of Common Forcing Functions

$\large f(t)$	$\large y_p(t)$
$\displaystyle {\large Ct^n} \quad \mbox{ (for } n = 0, 1, 2, \dots \mbox{)}$	$\displaystyle \large A_nt^n + A_{n-1}t^{n-1}+\ldots + A_0$
$\displaystyle \large Ce^{rt}$	$\displaystyle \large Ae^{rt}$
$\displaystyle \large Ce^{\alpha t} \cos{(\beta t)}$	$\displaystyle \large A e^{\alpha t}\cos{(\beta t)} + B e^{\alpha t}\sin{(\beta t)}$
$\displaystyle \large Ce^{\alpha t} \sin{(\beta t)}$	$\displaystyle \large Ae^{\alpha t} \cos{(\beta t)} + B e^{\alpha t}\sin{(\beta t)}$

Identifying and Adjusting for Rseonance

You have resonance when your initial guess for the particular solution is a homogeneous solution.
When there is resonance, multiply the guess by a multiple of $t$ (until the guess is no longer a homogeneous solution).
After adjusting your guess for $y_p(t)$, plug into the differential equation and solve for the undetermined coefficients.

Finding the General Solution to the Nonhomogeneous Case

Consider a nonhomogeneous differential equation of the form

\[ a y''+by'+cy=f(t). \]

Let $y_h(t)$ denote a general solution to corresponding homogeneous equation $a y''+by'+cy=0$, and let $y_p(t)$ denote a particular solution for the nonhomogeneous equation (when $f(t) \ne 0$).

Let’s explore how we can use both of these solutions to construct a general solution to nonhomogeneous differential equation.

Question 1:

Let $y_h(t)$ and $y_p(t)$ denote homogeneous and particular solutions, respectively, to the differential equation $ay''+by'+cy=f(t)$. Show that their sum $y(t)=y_h(t)+y_p(t)$ is a general solution to the nonhomogeneous differential equation.

Solution to Question 1:

Question 2:

Consider the differential equation \[ \frac{d^2x}{dt^2}-\frac{dx}{dt}-12x=e^{4t}.\]

Question 2a:

Find a general solution to the homogeneous equation.

Solution to Question 2a:

Question 2b:

Find a particular solution to the nonhomogeneous differential equation.

Solution to Question 2b:

Question 2c:

Use your previous answers to give a general solution to nonhomogeneous differential equation.

Solution to Question 2c:

Sums of Different Types of Forcing Functions

To find the general solution to a nonhomogeneous differential equation you simply add a particular solution to a general solution to the corresponding homogeneous equation. This 3-step strategy:

Find a general solution to the corresponding homogeneous equation
Find a particular solution for each part of the nonhomgeneous forcing function.
Add the previous results.

This process is often called the Method of Undetermined Coefficients.

Question 3:

Let $y_1(t)$ denote a particular solution to $ay''+by'+cy=f_1(t)$ and $y_2(t)$ denote a particular solution to $ay''+by'+cy=f_2(t)$. Show that $y_p(t) = y_1(t) + y_2(t)$ is a particular solution to $ay''+by'+cy=f_1(t)+f_2(t)$.

Solution to Question 3:

Question 4:

Consider the differential equation

\[ \frac{d^2x}{dt^2}+10\frac{dx}{dt}+9x=85\sin(2t)+18.\]

Recall that we have already worked on parts of this question. We have found that $x_h (t) = C_1e^{-9t}+C_2e^{-t}$ and the particular solution corresponding to $f_1(t) = 85\sin(2t)$ is $x_{1}(t) = -4\cos{(2t)}+\sin{(2t)}$. Using these results, finish solving the differential equation and give the general solution.

Solution to Question 4:

Products of Different Types of Forcing Functions

We can use the method of undetermined coefficients when the forcing function $f(t)$ is:

A power function, $f(t) = Ct^n$ (for $n$ a nonnegative integer).
A function of the form $f(t) = Ce^{\alpha t}\cos{(\beta t)}$ or $f(t) = Ce^{\alpha t}\sin{(\beta t)}$.

Product of Power and Exponential Functions

For a product of a power function $t^n$ and an exponential such as $f(t) = Ct^ne^{rt}$, we guess

\[\color{dodgerblue}{ y_p(t) = \big( A_n t^n + A_{n_1} t^{n-1} + \ldots A_1 t + A_0 \big) e^{rt} }.\]

Multiply by $t$ if there is resonance ($r$ is a real root (not repeated) of the the characteristic equation.)
Multiply by $t^2$ if there is still resonance ($r$ is a repeated real root of the the characteristic equation).

Question 5:

Give a guess for the particular solution to \[ y'' - 5y' -6y = 4t^2e^{6t}.\]

Solution to Question 5:

Products of Power, Exponential, and Sine (or Cosine) Functions

For a product of a power function, an exponential, and either a sine or cosine such as

$f(t) = Ct^ne^{\alpha t}\cos{(\beta t)}$ or $f(t) = Ct^ne^{\alpha t}\sin{(\beta t)}$, a similar strategy works if our guess has the form:

\[\color{dodgerblue}{y_p(t) = \big( A_n t^n + A_{n-1} t^{n-1} + \ldots A_1 t + A_0 \big) e^{\alpha t}\cos{(\beta t)} + \big( B_n t^n + B_{n-1} t^{n-1} + \ldots B_1 t + B_0 \big) e^{\alpha t}\sin{(\beta t)}}.\]

Multiply by $t$ if there is resonance (if $\alpha \pm i \beta$ are the complex roots of the the characteristic equation).

Question 6:

Give a guess for the particular solution to \[ y'' - 5y' -6y = 4t^2e^{6t}\sin{t}.\]

Solution to Question 6:

Question 7:

Consider the differential equation $x''-8x'+12x=f(t)$. For each $f(t)$, what would be your guess for the particular solution? Do not solve for the undetermined coefficients. If the method of undetermined coefficients cannot be applied, explain why not.

Question 7a:

$f(t) = 10\sin{(2t)}$

Solution to Question 7a:

Question 7b:

$f(t) = 10e^{6t}\sin{(2t)}$

Solution to Question 7b:

Question 7c:

$f(t) = 10\tan{(2t)}$

Solution to Question 7c:

Question 7d:

$f(t) = 10te^{2t}$

Solution to Question 7d:

Question 7e:

$f(t) = 8t^{-2}$

Solution to Question 7e:

Creative Commons License Information

Exploring Differential Equations by Adam Spiegler is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Based on a work at https://github.com/CU-Denver-MathStats-OER/ODEs and original content created by Rasmussen, C., Keene, K. A., Dunmyre, J., & Fortune, N. (2018). Inquiry oriented differential equations: Course materials. Available at https://iode.sdsu.edu.

\(\large f(t)\)	\(\large y_p(t)\)
\(\displaystyle {\large Ct^n} \quad \mbox{ (for } n = 0, 1, 2, \dots \mbox{)}\)	\(\displaystyle \large A_nt^n + A_{n-1}t^{n-1}+\ldots + A_0\)
\(\displaystyle \large Ce^{rt}\)	\(\displaystyle \large Ae^{rt}\)
\(\displaystyle \large Ce^{\alpha t} \cos{(\beta t)}\)	\(\displaystyle \large A e^{\alpha t}\cos{(\beta t)} + B e^{\alpha t}\sin{(\beta t)}\)
\(\displaystyle \large Ce^{\alpha t} \sin{(\beta t)}\)	\(\displaystyle \large Ae^{\alpha t} \cos{(\beta t)} + B e^{\alpha t}\sin{(\beta t)}\)