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Particular Solution Of The Differential Equation

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.

1.	What Is a Particular Solution Of The Differential Equation?
2.	How To Find Particular Solution Of Differential Equation?
3.	Particular Solution Vs General Solution Of Differential Equation
4.	Examples on Particular Solution Of The Differential Equation
5.	Practice Questions
6.	FAQs On Particular 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.

Related Topics

The following topics will help in a better understanding of the particular solution of the differential equation.

Examples on Particular Solution Of The Differential Equation

Example 1: Find if the equation y = e^-2xis a particular solution of a differential equation d²y/dx² + dy/dx -2y = 0.

Solution:

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^-2xis 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?

Solution:

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.

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Practice Questions on Particular Solution Of The Differential Equation

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FAQs on Particular Solution Of The Differential Equation

What Is The Particular Solution Of The Differential Equation?

A particular solution of differential equation is a solution of the form y = f(x), which do not have any arbitrary constants. The general solution of the differential equation is of the form y = f(x) or y = ax + b and it has a, b as its arbitrary constants. Attributing values to these arbitrary constants results in the particular solutions such as y = 2x + 1, y = 3x + 4, y = 5x + 2.

What Is The Difference Between General Solution And Particular Solution Of Differential Equation?

The particular solutions have been derived from the general solutions. The general solution of a differential equation has arbitrary constants, and the solutions without any arbitrary constants is called the particular solution of the differential equation. The general solution of the differential equation represents a family of curves or a set of parallel lines, and each of these lines or curves can be terms as the particular solution of differential equation.

How Do You Identify A Particular Solution Of Differential Equation?

The particular solution of differential equation can be easily identified, as it does not have any arbitrary constants. The solutions y = 3x + 3, y = x² + 11x + 7, are the examples of particular solution of differential equation.

What Is The Use Of a Particular Solution Of Differential Equation?

The particular solution of differential equation is useful to find the exact solution satisfying the differential equation, at a particular point or for a particular value of the independent variable.

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