Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20
Use the method of variation of parameters to find a particular solution of the
given differential equation. Then check your answer by using the method of
undetermined coefficients.
Short Answer
Expert verified
Question: Find the particular solution to the given second-order differential equation using variation of parameters and undetermined coefficients methods:
Answer: The particular solution is:
Step by step solution
01
Find the characteristic equation and roots
The characteristic equation of this homogeneous equation is:
This quadratic equation can be factored as follows:
So, the roots are r = 2, 3.
02
Write the complementary solution
Our complementary solution is a linear combination of our root solutions:
Now, let's move on to finding the particular solution using the variation of parameters method.
03
Find the Wronskian
The Wronskian, W, is used to find the coefficients of our particular solution using the variation of parameters formula:
04
Calculate u_1 and u_2
Using the variation of parameters formulas, we'll integrate to find u_1 and u_2:
Since we are looking for the particular solution, the constants of integration can be set to 0:
05
Form the particular solution
Now, we can write our particular solution y_p using u_1 and u_2:
Now, let's verify this solution using the method of undetermined coefficients.
06
Choose the form of particular solution using undetermined coefficients
The forcing function suggests that the particular solution must have the form:
07
Differentiate and substitute into the inhomogeneous equation
We need to differentiate and substitute it into the inhomogeneous equation:
Substitute the terms into the inhomogeneous equation:
08
Solve for A
Now, we equate coefficients to solve for A:
So our particular solution using undetermined coefficients is:
Since we've arrived at the same answer for our particular solution using both methods, we have successfully checked our answer. Our final solution for the given differential equation is:
Over 30 million students worldwide already upgrade their
learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Characteristic Equation
The characteristic equation is a crucial part of solving linear differential equations, especially those with constant coefficients. In this context, it helps find the roots that are used to form the complementary solution. For our differential equation, the focus was on the homogeneous part, given as To find the characteristic equation, we substitute possible solutions of the form into the equation. This results in a characteristic equation of the form:Using the quadratic formula or factoring, we find the roots: and . These roots indicate the exponents in the expression for the complementary solution. They also provide the fundamental solutions that represent the homogeneous system. The complementary solution, , derived from these roots is:
Undetermined Coefficients
The method of undetermined coefficients is a powerful tool used to find particular solutions for non-homogeneous linear differential equations. This method is especially effective for differential equations where the non-homogeneous part (the forcing function) is a simple polynomial, sinusoidal, or exponential function. For instance, in our example, the non-homogeneous term is . By the method of undetermined coefficients, we assume a particular solution of the form , where is an unknown coefficient to be determined.The key steps include:
Choosing a trial solution that matches the form of the non-homogeneous term.
Deriving the derivatives of the trial function.
Substituting these derivatives into the original differential equation.
Solving for the unknown coefficients by matching terms.
In our case, this results in , giving a particular solution:
Wronskian
The Wronskian is a determinant used in the theory of differential equations to establish whether a set of solutions is linearly independent. In the variation of parameters method, it plays a pivotal role in constructing the particular solution for a linear differential equation.To calculate the Wronskian for the given fundamental solutions and , we set up the Wronskian determinant as follows:Upon calculating, we findThis determination is crucial because it divides elements within the integration process when you calculate functions and necessary for the particular solution in variation of parameters. It ensures the independence of the solutions involved and assures that our particular solution is valid.
Particular Solution
Finding a particular solution for a non-homogeneous differential equation is an essential step that pinpoints the specific function which satisfies the entire equation. The method of variation of parameters is used here precisely due to its general applicability.For the differential equationwe seek a function whose derivatives satisfy the whole equation. We adopt a formula based on the Wronskian and integrals, which involve the functions and . These functions are solved from:
In this context, is the non-homogeneous part , , and .After integration:
Hence, the particular solution becomes:This particular solution fits when combined with the complementary solution, providing a comprehensive solution to the differential equation.
One App. One Place for Learning.
All the tools & learning materials you need for study success - in one app.
This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept
Privacy & Cookies Policy
Privacy Overview
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the ...
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.
Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.