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 the variation of parameters method in the context of CBSE Class 12 Maths differential equations?
The variation of parameters is a technique used to find a particular solution for non-homogeneous linear differential equations, specifically those of the form y'' + P(x)y' + Q(x)y = f(x). This method replaces arbitrary constants in the complementary function with functions and solves for them to satisfy the non-homogeneous equation, making it flexible for various types of right-hand side functions.
2. How do you solve a second-order linear differential equation using the variation of parameters method according to CBSE guidelines?
To solve using the variation of parameters method:
- First, find two linearly independent solutions (y1 and y2) of the corresponding homogeneous equation.
- Assume a particular solution of the form yp = u1(x)y1(x) + u2(x)y2(x), where u1 and u2 are functions to determine.
- Set up equations for u1' and u2' using the original equation and solve them via integration.
- Add the complementary function (from the homogeneous equation) to the particular solution to get the general solution.
3. When should the method of undetermined coefficients be preferred over variation of parameters?
The method of undetermined coefficients is preferred when the non-homogeneous term f(x) is a polynomial, exponential, or trigonometric function (or their combinations). It is usually faster and involves simpler calculations compared to the variation of parameters when these conditions are met.
4. Why is it necessary for the solutions y1(x) and y2(x) to be linearly independent in the variation of parameters method?
Linearly independent solutions ensure that the general solution of the homogeneous equation spans the entire solution space. If y1 and y2 are not independent, you cannot construct a valid particular solution using variation of parameters, leading to incomplete or incorrect answers.
5. What are the main conceptual errors students should avoid when applying variation of parameters in board exams?
Common errors include:
- Not checking for linear independence of y1 and y2.
- Setting up equations for u1 and u2 incorrectly.
- Missing integration constants or applying limits improperly.
- Forgetting to combine both particular and complementary functions for the final solution.
6. How does the Wronskian help in the variation of parameters method as practiced in Class 12 Maths?
The Wronskian is used to check the linear independence of y1 and y2. In solving for u1(x) and u2(x), the Wronskian appears in the denominators. A non-zero Wronskian verifies that your chosen solutions are valid and suitable for the method.
7. Can variation of parameters be extended to higher-order differential equations beyond those in the CBSE 2025–26 curriculum?
Yes, the variation of parameters method applies to linear non-homogeneous differential equations of any order. However, for CBSE Class 12, the syllabus focuses on second-order cases, as higher-order equations involve more complex calculations.
8. Why is practicing solved examples using variation of parameters important for CBSE board preparation?
Practicing solved examples enhances understanding of integration techniques, stepwise logical solving, and helps students familiarize themselves with the structure of differential equation solutions. It prepares students for typical and advanced problems in exams, reducing mistakes and building confidence.
9. How is the general solution of a non-homogeneous differential equation structured when using the variation of parameters technique?
The general solution combines the complementary function (general solution of the homogeneous equation) and the particular integral (specific solution found using variation of parameters). It takes the form: General Solution = Complementary Function + Particular Integral.
10. What are common misconceptions about the scope of variation of parameters among Class 12 Maths students?
A common misconception is that variation of parameters can only handle complex functions. In reality, it works for any non-homogeneous linear equation but tends to be used when f(x) is not covered by the undetermined coefficients method. Another misconception is that integration is always difficult; with proper selection of y1 and y2, integrations can sometimes be straightforward.
