Lesson 8.4: Second-order Equations: Particular Integrals and Applications

Introduction

Welcome to Lesson 8.4 of Foundation Further Mathematics! In this lesson, we will explore second-order differential equations and how to find their particular integrals.

Learning Objectives

By the end of this lesson, you should be able to:

• Find a particular integral for standard forcing terms like polynomial, exponential, and trigonometric functions.
• Understand that the complete solution of a second-order differential equation is the sum of the complementary function and the particular integral.
• Apply boundary conditions to your solutions and model physical systems that are forced and/or damped.
• Find a particular integral for a given right-hand side.
• Write down and apply the complete solution.

Understanding Second-order Linear Differential Equations

A second-order linear differential equation can be represented in the standard form:

$$\frac{d^2y}{dt^2} + p(t)\frac{dy}{dt} + q(t)y = g(t)$$

where:

• $y$ is the unknown function of $t$.
• $p(t)$ and $q(t)$ are continuous functions.
• $g(t)$ is the forcing function (right-hand side).

The general solution of such equations consists of two parts:

1. The complementary function ($y_c$), which is the solution to the associated homogeneous equation.
2. The particular integral ($y_p$), which is a specific solution to the non-homogeneous equation.

Find the Complementary Function

Before finding the particular integral, you must first find the complementary function. To solve the homogeneous equation:

$$\frac{d^2y}{dt^2} + p(t)\frac{dy}{dt} + q(t)y = 0$$

You typically assume a solution of the form:

$$y = e^{rt}$$

Substituting this into the homogeneous part leads to the characteristic equation:

$$r^2 + pr + q = 0$$

This can be solved to find the roots $r_1$ and $r_2$, which help establish the form of the complementary function:

• If the roots are distinct, then:

$$y_c = C_1 e^{r_1 t} + C_2 e^{r_2 t}$$

• If the roots are repeated, then:

$$y_c = (C_1 + C_2 t)e^{r t}$$

Finding Particular Integrals

To find the particular integral $y_p$, we follow certain methods depending on the form of the forcing function $g(t)$. Below are the common cases:

1. Polynomial Functions

If $g(t)$ is a polynomial of degree $n$, assume:

$$y_p = A_n t^n + A_{n-1} t^{n-1} + ... + A_0$$

where $A_n, A_{n-1}, ..., A_0$ are constants to be determined.

Example: For the equation:

$$\frac{d^2y}{dt^2} - 3\frac{dy}{dt} + 2y = 6t^2$$

Assume:

$$y_p = At^2 + Bt + C$$

Plugging this expression into the left-hand side will allow you to find $A$, $B$, and $C$.

2. Exponential Functions

If $g(t)$ is an exponential function like $g(t) = e^{kt}$, then:

$$y_p = Ae^{kt}$$

Example: For:

$$\frac{d^2y}{dt^2} - 2\frac{dy}{dt} + y = e^{3t}$$

Try:

$$y_p = Ae^{3t}$$

3. Trigonometric Functions

If $g(t)$ is a trigonometric function like $g(t) = \sin(kt)$ or $g(t) = \cos(kt)$, assuming:

$$y_p = A\sin(kt) + B\cos(kt)$$

is often effective.

Example: For:

$$\frac{d^2y}{dt^2} + y = \sin(4t)$$

Then:

$$y_p = A\sin(4t) + B\cos(4t)$$

Constructing the Complete Solution

Once you have both the complementary function and particular integral, the complete solution can be expressed as:

$$y = y_c + y_p$$

This combined solution allows you to model a variety of physical scenarios, such as damped oscillations or forced vibrations.

Applying Boundary Conditions

To find specific constants in your solution, you may need to apply boundary conditions. What this means is that you will plug in specific values of $t$ and $y$ to determine constants like $C_1$, $C_2$, and coefficients from your particular integral.

Example: If you had:

$$y(0) = 3$$

and $\frac{dy}{dt}(0) = 1$, plug these into your complete solution to find an explicit formula.

Conclusion

In this lesson, we learned how to tackle second-order differential equations, focusing on finding particular integrals for various forcing functions. By combining the complementary function with the particular integral, we can effectively solve real-world problems modeled by these equations.

Study Notes

• The general form of a second-order linear equation is $\frac{d^2y}{dt^2} + p(t)\frac{dy}{dt} + q(t)y = g(t)$.
• The solution consists of the complementary function ($y_c$) and particular integral ($y_p$).
• For polynomial forcing functions, assume $y_p$ as a polynomial of the same degree.
• For exponential forcing functions, assume $y_p$ as a function of the form $y_p = Ae^{kt}$.
• For trigonometric forcing functions, assume $y_p$ as a combination of sine and cosine functions.
• Apply boundary conditions to determine specific constants in your solutions.