How to Solve Differential Equations Using Variation of Parameters
The Variation in Maths is mostly based on the maths of Class 12 Mathematics which the students need to learn for the chapter named differential equations. This concept is used in many competitive examinations for students pursuing fields related to mathematics. The students will learn about what is meant by the Variation of Parameters. Then, the notes provide the student with a better glimpse by explaining the method with a generic example with step-by-step calculation. Then, you get to understand the method in Variation of parameters and 2 methods in Variation of Parameters. The end is concluded with a solved example so you get to solve it first and then correct your mistakes if there have been any.
What is Variation in Maths?
Variation of parameters or let’s say variation in mathematics is a general method of finding a specific solution of a differential equation through replacing the constants in the solution of an associated (homogeneous) equation by functions and identifying these functions such that the original differential equation is satisfied.
In order to illustrate the method, let’s say it is desired to determine a specific solution of the equation: - y″ + p(x) y′ + q(x) y = g(x).
Variation of Parameters
In order to illustrate the method, let’s say it is desired to determine a specific solution of the equation: - y″ + p(x) y′ + q(x) y = g(x).
For the purpose of using this method, it is necessary to first know the general solution of the corresponding homogeneous equation—i.e., an associated equation where the right-hand side is zero. If y1(x) and y2(x) are two different solutions of the equation, then any combination ay1(x) + by2(x) is also a solution, known as the general solution, for any constants a and b.
The variation of parameters involves replacing the constants a and b by functions u1(x) and u2(x) and identifying what these functions must be to satiate the actual non-homogeneous equation. After a few manipulations, it can be presented that if the functions u1(x) and u2(x) satisfies the mathematical expression u′1y1 + u′2y2 = 0 and u1′y1′ + u2′y2′ = g, then u1y1 + u2y2 will satiate the original differential equation. The last two equations can be solved for providing the u1′ = −y2g/(y1y2′ − y1′y2) and u2′ = y1g/(y1y2′ − y1′y2). These last equations either will identify u1 and u2 or else will cater as an initial point for determining an estimated solution.
Method of Variation of Parameters
In this lesson, we will discuss the method of variation of parameters with respect to second-order differential equations of this type:
\[\frac {D^2 y}{dx^2} + P(x) \frac {dy}{dx} + Q(x)y = f(x)\]
where P(x), Q(x) and f(x) are said to be functions of x.
Two Methods in Variation of Parameters
There are mainly 2 methods of solving equations like:
\[\frac {D^2 y}{dx^2} + P(x) \frac {dy}{dx} + Q(x)y = f(x)\]
Undetermined Coefficients that only work when f(x) will be a polynomial, exponential,
sine, cosine, or a linear combination of those.
Variation of Parameters works on a wide range of functions but is a bit messy to use.
Solutions to Variation of Parameters
In order to keep things simple, we will only look at the case: d2y
\[Dx^2 + P \frac {dy}{dx} + qy = f(x)\]
In which, p and q are constants and f(x) is a non-zero function of x.
A full-fledged solution to such an equation can be identified by combining two types of solution i.e.:
The general solution of the homogeneous equation expressed as \[\frac{d^2 y}{dx^2} + P \frac {dy}{dx} + qy = 0\]
Particular solutions of the non-homogeneous equation expressed as \[\frac {d^2 y}{dx^2} + \frac {dy}{dx} + qy = f(x)\]
Remember that f(x) can be a single function or a sum of two or more functions.
Once we have determined the general solution and all the particular solutions, then the ultimate complete solution is identified by adding up all the solutions together.
This method depends upon integration.
A minor issue with this method is that, although it may produce a solution, in some cases the solution has to be left as an integral.
Solved Example using Variation of Parameter Formula
Example:
Solve the following equation: \[\frac {d^2 y}{dx^2} - 3 \frac {dy}{dx} + 2y = e^x\]
Finding the general solution of \[\frac {d^2 y}{dx^2} - 3 \frac {dy}{dx} + 2y = 0\]
Solution:
The characteristic equation will be: r2 − 3r + 2 = 0
Factor: (r − 1)(r − 2) = 0
r = 1 or 2
Thus, the general solution of the differential equation is y = Aex + Be2x
Therefore, in this case, the fundamental solutions and their derivatives will be:
y1(x) = ex
y1'(x) = ex
y2(x) = e2x
y2'(x) = 2e2x
FAQs on Variation of Parameters: Complete Guide for Students
1. What is variation of parameters in differential equations?
The variation of parameters is a method used to find a particular solution to a nonhomogeneous linear differential equation. It involves allowing the constants in the complementary solution to vary into functions, helping solve equations where the right side is not zero.
2. When do you use the variation of parameters method?
You use variation of parameters when solving a nonhomogeneous linear differential equation, especially if the nonhomogeneous term does not fit methods like undetermined coefficients. This approach works for equations with variable or complicated right-hand sides.
3. How does the variation of parameters formula work?
The formula for variation of parameters involves finding two functions, $u_1(x)$ and $u_2(x)$, such that the particular solution is $y_p = u_1(x) y_1(x) + u_2(x) y_2(x)$.
- $y_1(x)$, $y_2(x)$ are solutions to the homogeneous equation.
4. What is the general solution using variation of parameters?
The general solution to a nonhomogeneous linear equation using variation of parameters combines the complementary (homogeneous) solution and a particular solution: $y(x) = y_c(x) + y_p(x)$, where $y_p(x)$ is found using the method of variation of parameters.
5. What are the steps in applying variation of parameters for second-order differential equations?
To use variation of parameters for second-order equations:
- Solve the homogeneous equation for $y_1(x)$ and $y_2(x)$.
- Set up $y_p = u_1(x) y_1(x) + u_2(x) y_2(x)$.
- Find $u_1(x)$ and $u_2(x)$ using integrals.
- Add the particular solution to the complementary solution.
6. Can you use variation of parameters for higher-order differential equations?
Yes, the variation of parameters method can be extended to higher-order linear differential equations. However, the calculations become more complicated because you need more complementary solutions and functions to solve for, usually involving systems of equations and multiple integrals.
7. What are the advantages of using variation of parameters?
Variation of parameters is a versatile technique because:
- It works with any nonhomogeneous term, not just special forms.
- It applies to equations with variable coefficients.
- It provides a general process when other methods fail.
8. What is the limitation of the variation of parameters method?
A main limitation of variation of parameters is that the process can be algebraically intensive. It often requires complex integration, which may be difficult or impossible to simplify for certain nonhomogeneous terms, making the method less practical in some cases.
9. How are integrals used in variation of parameters?
In variation of parameters, integrals are used to find the unknown functions $u_1(x)$ and $u_2(x)$ that multiply the complementary solutions. These functions are determined by integrating expressions involving the nonhomogeneous term and the Wronskian of the homogeneous solutions.
10. Why is the Wronskian important in variation of parameters?
The Wronskian is a determinant used to check if solutions to the homogeneous equation are linearly independent. In variation of parameters, it appears in the denominators when integrating to find the particular solution, ensuring the solutions you use are valid.
11. How does variation of parameters differ from undetermined coefficients?
Unlike undetermined coefficients, which only works for simple forms like polynomials or exponentials, variation of parameters does not limit the form of the nonhomogeneous term. It uses integrals to find solutions, offering broader application for complex equations.
12. What types of equations benefit most from variation of parameters?
Equations with nonhomogeneous terms that are not simple polynomials, exponentials, or trigonometric functions benefit most. This includes equations with variable coefficients or complicated right-hand sides, making variation of parameters a powerful and flexible method.
