Problem 129 - Calculus 3

Question

For the following problems, find the general solution. $$y^{\prime \prime}+5 y+y=x+e^{2 x}$$

Step-by-Step Solution

Answer

The general solution is \( y(x) = C_1\cos{(\sqrt{6}x)} + C_2\sin{(\sqrt{6}x)} + \frac{1}{6}x + \frac{1}{20}e^{2x} \).

Step 1: Identify the Type of Equation

The given differential equation is \( y'' + 5y + y = x + e^{2x} \), which simplifies to \( y'' + 6y = x + e^{2x} \). This is a non-homogeneous linear differential equation.

Step 2: Solve the Homogeneous Equation

The associated homogeneous equation is \( y'' + 6y = 0 \). Solving the characteristic equation \( r^2 + 6 = 0 \), we find roots \( r = \pm i \sqrt{6} \), giving the complementary solution: \( y_c = C_1\cos{(\sqrt{6}x)} + C_2\sin{(\sqrt{6}x)} \).

Step 3: Find the Particular Solution

To solve \( y'' + 6y = x + e^{2x} \), we use the method of undetermined coefficients. We assume a particular solution of the form \( y_p = Ax + B + Ce^{2x} \).

Step 4: Substitute and Solve for Coefficients

Substitute \( y_p = Ax + B + Ce^{2x} \) into the original equation \( y'' + 6y = x + e^{2x} \). Solving for \( A, B, \) and \( C \) gives \( A = \frac{1}{6} \), \( B = 0 \), and \( C = \frac{1}{20} \). The particular solution is \( y_p = \frac{1}{6}x + \frac{1}{20}e^{2x} \).

Step 5: Write the General Solution

Combine the complementary solution and the particular solution to write the general solution: \( y(x) = C_1\cos{(\sqrt{6}x)} + C_2\sin{(\sqrt{6}x)} + \frac{1}{6}x + \frac{1}{20}e^{2x} \).

Key Concepts

Non-Homogeneous Differential Equation

A non-homogeneous differential equation can be thought of as a mathematical equation that expresses a relationship involving a function and its derivatives. However, unlike homogeneous equations that equal zero, a non-homogeneous equation has a non-zero term or function on the right side. In simple terms, it resembles a typical equation but with an extra part.

For example, in the given problem, the equation was originally expressed as \( y'' + 5y + y = x + e^{2x} \) and then simplified to \( y'' + 6y = x + e^{2x} \). The right-hand side contains \( x + e^{2x} \), which makes it non-homogeneous. This additional term could be anything from a simple variable expression to complex functions that aren't inherently zero.

• Non-zero terms signify an external influence or force affecting the system represented by the differential equation.
• Solving non-homogeneous equations typically involves special techniques to account for these non-zero terms, such as determining the complementary and particular solutions.

Understanding the concept of non-homogeneity helps in determining the approach needed to find the complete solution to such equations.

Method of Undetermined Coefficients

The method of undetermined coefficients is a neat trick used to find a particular solution to non-homogeneous differential equations. It's particularly useful when the non-homogeneous part (the right side of the equation) is a simple, well-behaved function such as polynomials, exponentials, or trigonometric functions.

Essentially, this method involves assuming a form for the particular solution based on the type of function on the non-homogeneous side. Then, we solve for unknown constants within this assumed form. It's almost like making an educated guess!

In the given problem, we had the non-homogeneous part \( x + e^{2x} \). We assumed a particular solution of the form \( y_p = Ax + B + Ce^{2x} \).

• This form covered the linear function \( x \) and the exponential function \( e^{2x} \) separately.
• Once you have your assumed solution form, you substitute it back into the original differential equation.
• You then solve for the coefficients (A, B, C) by matching terms on both sides of the equation, giving us \( A = \frac{1}{6}, B = 0, \) and \( C = \frac{1}{20} \).

The method is straightforward once the correct form of the particular solution is identified!

Complementary Solution

The complementary solution is a critical part of solving a differential equation, particularly the associated homogeneous equation. This equation is derived by removing the non-homogeneous part, leaving a simpler homogeneous counterpart.

For example, in our problem, the associated homogeneous equation was \( y'' + 6y = 0 \). Here's a quick breakdown of how we solve it:

• We start by finding the characteristic equation, which in this case is \( r^2 + 6 = 0 \).
• Solving this gives us roots that are imaginary numbers: \( r = \pm i \sqrt{6} \).
• Imaginary roots indicate that the complementary solution will involve trigonometric terms. Thus, the complementary solution results in \( y_c = C_1 \cos{(\sqrt{6}x)} + C_2 \sin{(\sqrt{6}x)} \).

The complementary solution accounts for the natural behavior of the system described by the differential equation absent any external influences. By determining both the complementary and particular solutions, we are able to write the full general solution to the non-homogeneous differential equation.