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General and Particular Solution of a Differential Equation | Mathematics for NDA PDF Download

Introduction

Particular solution of the differential equation is a unique solution of the form y = f(x), which satisfies the differential equation. The particular solution of the differential equation is derived by assigning values to the arbitrary constants of the general solution of the differential equation.

Let us learn more about the particular solution of the differential equation, how to find the solution of the differential equation, and the difference between a particular solution and the general solution of the differential equation.

What is Particular Solution of the Differential Equation?

Particular solution of the differential equation is an equation of the form y = f(x), which do not contain any arbitrary constants, and it satisfies the differential equation. The equation or a function of the form y = f(x), having specific values of x which satisfy this equation and are called the solutions of this equation. For a differential equation d2y/dx2 + 2dy/dx + y = 0, the the values of y which satisfy this differential equation is called the solution of the differential equation.

Here y = f(x) representing a line or a curve is the solution of the differential equation that satisfies the differential equation. The solution of the form y = ax2 + bx + c is the general solution of the differential equation, since it contains arbitrary constants a, b, c. Further, if the solution has values assigned to these arbitrary constants, or if the solution is without any arbitrary constants, then the solution is called the particular solution of the differential equation.

How to Find Particular Solution of Differential Equation?

The particular solution of the differential equation can be computed from the general solution of the differential equation. The general solution of a differential solution would be of the form y = f(x) which could be any of the parallel line or a curve, and by identifying a point that satisfies one of these lines or curves, we can find the exact equation of the form y = f(x) which is the particular solution of the differential equation.

The following steps help in finding the particular solution of the differential equation.

  • The given differential equation is solved by separating the variables and integrating on both sides to obtain the general solution of the differential equation.
  • For differential equations that cannot be solved easily, different methods are employed to find the general solution of the differential equation.
  • The general solution of the differential equation contains arbitrary constants, which have to be assigned suitable values to get a particular solution of the differential equation.
  • A point is identified to help substitute the values for the arbitrary constants, to obtain the particular solution of the differential equation.
  • There can be more than one particular solution for the differential equation, based on the different values of the arbitrary constant.

Particular Solution vs General Solution of Differential Equation

A particular solution of the differential equation is derived from the general solution of the differential equation. The differential equation has one general solution, and numerous particular solutions, based on the different values of the arbitrary constants of the general solution.

The general solution of the differential represents a family of curves or lines in the coordinate plane, These curves or lines represent a set of parallel lines or curves, and each of these lines or the curves can be identified as the particular solution of the differential equation.

The general solution of the differential equation is of the form y = ax + b, but the particular solution of the differential equation can be y = 3x + 4, y = 5x + 7, y = 2x + 1. These particular solutions of the differential equation have been obtained by assigning different values to the arbitrary constants a, b in the general solution of the differential equation.

Solved Example

Example 1: Find if the equation y = e-2x is a particular solution of a differential equation d2y/dx2 + dy/dx -2y = 0.
Sol:
The given equation of the solution of the differential equation is y = e-2x.
Differentiating this above solution equation on both sides we have the following expression.
dy/dx = -2e-2x
Further, differentiating this with respect to x for the second differentiation, we have:
d2y/dx2 = 4e-2x
Applying this in the differential equation to check if it satisfies the given expression.
The given differential equation is:
d2y/dx2 + dy/dx -2y = 0
4e-2x - 2e-2x - 2e-2x = 0.
Therefore, the equation y = e-2x is a solution of a differential equation d2y/dx2 + dy/dx -2y = 0.

Example 2: Verify if the function y = acosx + bsinx is a particular solution of a differential equation y'' + y = 0?
Sol:
The given function is y = aCosx + bSinx.
Let us take the second derivative of this function.
y' = -aSinx + bCosx
y'' = -aCosx - bSinx
Further we can substitute this second derivative value in the below differential equation.
y'' + y = 0
(-aCosx - bSinx) + (aCosx + bSinx.) = 0
-aCosx - bSinx + aCosx + bSinx. = 0
-aCosx + aCosx -bSinx +bSinx = 0
Therefore, the function y = acosx + bsinx is a solution of a differential equation y'' + y = 0.

The document General and Particular Solution of a Differential Equation | Mathematics for NDA is a part of the NDA Course Mathematics for NDA.
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FAQs on General and Particular Solution of a Differential Equation - Mathematics for NDA

1. What is a particular solution of a differential equation?
Ans. A particular solution of a differential equation is a solution that satisfies the differential equation along with given initial or boundary conditions. It is a specific solution that meets the specific criteria set by the problem at hand.
2. How can we find the particular solution of a differential equation?
Ans. To find the particular solution of a differential equation, we typically use the method of undetermined coefficients or variation of parameters. These methods involve solving the differential equation with the given initial or boundary conditions to determine the specific solution.
3. What is the difference between a particular solution and a general solution of a differential equation?
Ans. The general solution of a differential equation includes all possible solutions with arbitrary constants, while a particular solution is a specific solution that meets certain conditions. The general solution provides a family of solutions, whereas the particular solution is one unique solution within that family.
4. Can a differential equation have more than one particular solution?
Ans. No, a differential equation typically has only one particular solution that satisfies the given initial or boundary conditions. The particular solution is unique and specific to the conditions provided in the problem.
5. Why is finding the particular solution of a differential equation important?
Ans. Finding the particular solution of a differential equation is important as it allows us to determine the specific solution that meets the given conditions. This specific solution provides valuable information about the system or process being modeled by the differential equation and helps in making accurate predictions and calculations.
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Dec 23, 2024 Last updated
Document Description: General and Particular Solution of a Differential Equation for NDA 2024 is part of Mathematics for NDA preparation. The notes and questions for General and Particular Solution of a Differential Equation have been prepared according to the NDA exam syllabus. Information about General and Particular Solution of a Differential Equation covers topics like Introduction, What is Particular Solution of the Differential Equation?, How to Find Particular Solution of Differential Equation?, Particular Solution vs General Solution of Differential Equation, Solved Example and General and Particular Solution of a Differential Equation Example, for NDA 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for General and Particular Solution of a Differential Equation.
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