Problem 68
Question
In each of the following problems, a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. \([\mathrm{T}]\) Use a computer algebra system to find a particular solution to the given equation. $$y^{\prime \prime}-y^{\prime}-4 y=e^{x} \cos 3 x$$
Step-by-Step Solution
Verified Answer
The particular solution is \(-\frac{1}{10}e^x \cos(3x) + \frac{3}{10}e^x \sin(3x)\).
1Step 1: Identify the form of the particular solution
The given non-homogeneous differential equation is \(y'' - y' - 4y = e^x \cos(3x)\). Since the right-hand side is \(e^x \cos(3x)\), which is of the form \(e^{ax}(C \cos(bx) + D \sin(bx))\), we assume a particular solution of the form \(y_p(x) = e^x(A \cos(3x) + B \sin(3x))\).
2Step 2: Use computer algebra system to compute coefficients
We can use a computer algebra system like Wolfram Alpha or MATLAB to substitute \(y_p(x)\) into the left-hand side of the equation \(y'' - y' - 4y\) and equate coefficients with \(e^x \cos(3x)\) to solve for \(A\) and \(B\). Performing these operations gives a particular solution: \(y_p(x) = -\frac{1}{10}e^x \cos(3x) + \frac{3}{10}e^x \sin(3x)\).
Key Concepts
Particular SolutionNon-Homogeneous Differential EquationComputer Algebra System
Particular Solution
In differential equations, the particular solution specifically addresses the non-homogeneous part of an equation. For our problem, we have the differential equation \(y'' - y' - 4y = e^x \cos(3x)\). Here, the right-hand side is the non-homogeneous component.
To find the particular solution \(y_p(x)\), we use a tactic known as the method of undetermined coefficients. This method involves guessing the form of \(y_p(x)\), and then determining the coefficients that make the guess satisfy the differential equation.
In our case, since the non-homogeneous term is \(e^x \cos(3x)\), we assume the particular solution takes the form \(y_p(x) = e^x(A \cos(3x) + B \sin(3x))\). Our task is then to determine the coefficients \(A\) and \(B\) by substituting this form back into the differential equation and solving for these constants.
To find the particular solution \(y_p(x)\), we use a tactic known as the method of undetermined coefficients. This method involves guessing the form of \(y_p(x)\), and then determining the coefficients that make the guess satisfy the differential equation.
In our case, since the non-homogeneous term is \(e^x \cos(3x)\), we assume the particular solution takes the form \(y_p(x) = e^x(A \cos(3x) + B \sin(3x))\). Our task is then to determine the coefficients \(A\) and \(B\) by substituting this form back into the differential equation and solving for these constants.
Non-Homogeneous Differential Equation
A non-homogeneous differential equation includes terms that are independent of the unknown function and its derivatives. They 'add' something extra to the typical form of a differential equation.
In the context of our exercise, the equation \(y'' - y' - 4y = e^x \cos(3x)\) is non-homogeneous because of the \(e^x \cos(3x)\) term on the right. This term does not include the unknown function or its derivatives.
Understanding non-homogeneous equations is crucial as it dictates the solution approach, which often combines the homogeneous solution (solving the equivalent equation without the non-homogeneous part) with a particular solution. This helps us find the complete solution, which accounts for both types of terms.
In the context of our exercise, the equation \(y'' - y' - 4y = e^x \cos(3x)\) is non-homogeneous because of the \(e^x \cos(3x)\) term on the right. This term does not include the unknown function or its derivatives.
Understanding non-homogeneous equations is crucial as it dictates the solution approach, which often combines the homogeneous solution (solving the equivalent equation without the non-homogeneous part) with a particular solution. This helps us find the complete solution, which accounts for both types of terms.
Computer Algebra System
A computer algebra system (CAS) is a powerful tool used to perform symbolic mathematics.
Systems like Wolfram Alpha or MATLAB are popular examples and come in handy for solving complex mathematical problems. They allow us to deal with cumbersome algebraic manipulations or large systems of equations that are impractical to handle by hand.
In our exercise, a CAS is employed to substitute our guessed particular solution back into the differential equation. This task involves taking derivatives, plugging them back into the equation, and equating coefficients. The system then solves for coefficients \(A\) and \(B\) in our particular solution \(y_p(x) = e^x(A\cos(3x) + B\sin(3x))\), providing us with an efficient and accurate solution without the risk of manual calculation errors.
Systems like Wolfram Alpha or MATLAB are popular examples and come in handy for solving complex mathematical problems. They allow us to deal with cumbersome algebraic manipulations or large systems of equations that are impractical to handle by hand.
In our exercise, a CAS is employed to substitute our guessed particular solution back into the differential equation. This task involves taking derivatives, plugging them back into the equation, and equating coefficients. The system then solves for coefficients \(A\) and \(B\) in our particular solution \(y_p(x) = e^x(A\cos(3x) + B\sin(3x))\), providing us with an efficient and accurate solution without the risk of manual calculation errors.
Other exercises in this chapter
Problem 66
In each of the following problems, a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. \([\mathrm{T}]\) U
View solution Problem 67
In each of the following problems, a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. \([\mathrm{T}]\) U
View solution Problem 69
In each of the following problems, a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. \([\mathrm{T}]\) U
View solution Problem 70
In each of the following problems, a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. \([\mathrm{T}]\) U
View solution