Problem 31 - A First Course in Differential Equations with Modeling Applications

Question

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$\begin{aligned}&y^{\prime \prime}-7 y^{\prime}+10 y=24 e^{x}\\\&y=c_{1} e^{2 x}+c_{2} e^{5 x}+6 e^{x},(-\infty, \infty)\end{aligned}$$

Step-by-Step Solution

Answer

The given function is a general solution to the differential equation on \((-\infty, \infty)\).

Step 1: Identify the Differential Equation

The given differential equation is \( y'' - 7y' + 10y = 24e^x \). Our task is to verify that the function \( y = c_1 e^{2x} + c_2 e^{5x} + 6 e^x \) satisfies this equation on the interval \((-\infty, \infty)\).

Step 2: Differentiate the General Solution

Calculate the first and second derivatives of the given general solution.1. First derivative: \[ y' = 2c_1 e^{2x} + 5c_2 e^{5x} + 6 e^x \] 2. Second derivative: \[ y'' = 4c_1 e^{2x} + 25c_2 e^{5x} + 6 e^x \]

Step 3: Substitute into Differential Equation

Substitute \( y, y', \) and \( y'' \) into the differential equation:\[ y'' - 7y' + 10y = (4c_1 e^{2x} + 25c_2 e^{5x} + 6 e^x) - 7(2c_1 e^{2x} + 5c_2 e^{5x} + 6 e^x) + 10(c_1 e^{2x} + c_2 e^{5x} + 6 e^x) \]

Step 4: Simplify and Combine Like Terms

First, distribute in each term:\[4c_1 e^{2x} + 25c_2 e^{5x} + 6 e^x \]\[- 14c_1 e^{2x} - 35c_2 e^{5x} - 42 e^x \]\[ + 10c_1 e^{2x} + 10c_2 e^{5x} + 60 e^x \]Combine like terms for each exponential:- For \( e^{2x} \): \( 4c_1 - 14c_1 + 10c_1 = 0 \)- For \( e^{5x} \): \( 25c_2 - 35c_2 + 10c_2 = 0 \)- For \( e^x \): \( 6 - 42 + 60 = 24 \)

Step 5: Verify Result with Original Equation

The left side of the equation simplifies to \( 24e^x \).Since the equation simplifies to the nonhomogeneous term \( 24e^x \), the given function satisfies the differential equation fully.

Key Concepts

General Solution

In the context of differential equations, the general solution encompasses all possible solutions to a given equation. For second-order differential equations, such as our example, it involves combining the complementary solution (solution to the associated homogeneous equation) with a particular solution of the nonhomogeneous equation.

For our equation, the complementary or homogeneous part is determined by solving the equation without the external force term, in this case, without the \(24 e^x\). That results in solving: \[y'' - 7y' + 10y = 0\]Here, solutions are typically of the form \(e^{rx}\), which leads us to solve the characteristic polynomial obtained.

However, in this exercise, we are given the general solution already:\[y = c_{1} e^{2x} + c_{2} e^{5x} + 6 e^x\]This solution includes:

• The complementary solution: \(c_{1} e^{2x} + c_{2} e^{5x}\)
• The particular solution: \(6 e^x\)

These components, when combined, meet the requirements laid out by the nonhomogeneous differential equation for the entire real number line, \((-\infty, \infty)\).

Differential Equation Verification

Verification of a solution involves plugging the general solution and its derivatives back into the original differential equation to check it holds true. Here, it ensures that our given solution satisfies the entire equation:\[y'' - 7y' + 10y = 24 e^x\]To verify, compute the first and second derivatives of the proposed solution:

• First derivative: \(y' = 2c_1 e^{2x} + 5c_2 e^{5x} + 6 e^x\)
• Second derivative: \(y'' = 4c_1 e^{2x} + 25c_2 e^{5x} + 6 e^x\)

Substitute these into the left side of the equation and simplify the terms:\[(4c_1 e^{2x} + 25c_2 e^{5x} + 6 e^x) - 7(2c_1 e^{2x} + 5c_2 e^{5x} + 6 e^x) + 10(c_1 e^{2x} + c_2 e^{5x} + 6 e^x)\]Upon simplifying, terms relating to \(e^{2x}\) and \(e^{5x}\) cancel, and what remains should match the right-hand side of the equation, \(24e^x\). A match confirms that our function is indeed a solution, verifying the solution's validity over the specified range.

Second Order Differential Equations

Second order differential equations are those in which the highest derivative is a second derivative. They are expressed in the form:\[y'' + py' + qy = g(x)\]where \(p\) and \(q\) are constants or functions, and \(g(x)\) is often called the nonhomogeneous term.

These equations can describe a variety of physical systems, like harmonic oscillators or electrical circuits. Our exercise was to handle a particular type, called a linear nonhomogeneous second order differential equation.

To solve such an equation, one typically needs to:

• Find the general solution of the associated homogeneous equation (i.e., set \(g(x) = 0\))
• Identify a particular solution that specifically satisfies the nonhomogeneous part \(g(x)\)
• Combine both to form the general solution

The complexity of solutions can be further managed by using various methods like the method of undetermined coefficients or variation of parameters, frequently employed to handle the particular solution component when \(g(x)\) follows a specific pattern as in our exercise.