Solutions of A Differential Equation
Solutions of a differential equation is a normal equation of the curve y = f(x) which satisfies the differential equation. The differential equation has a general solution and a particular solution. The general solution contains arbitrary constants and assigning values to the arbitrary constants transforms it to a particular solution.
Let us learn more about how to find the solutions of a differential equation, and also to find the differential equation from the given solutions.
How To Find Solutions of A Differential Equation?
Solutions of a differential equation are the values or the equation or a curve, line which satisfy the given differential equation. A simple equation of the form x^{2} + 4 = 0, or Sin^{2}x + Cosx = 0, has solutions as numbers, real numbers or complex numbers which satisfy the simple equation. If the particular value is a solution of an equation, it can be substituted in place of x in the equation, and the lefthand side of the equation is equal to the righthand side of the equation.
Further let us consider the differential equation d^{2}y/dx^{2} + y = 0. Comparing this differential equation of the earlier two simple equations, the solution of a differential equation is a curve of the form y = f(x), which satisfy the differential equation Also if the solution y = f(x), if substituted in the solution of the differential equation, the left hand side of the equation is equal to the righthand side of the equation.
Particular Solutions And General Solution of a Differential Equation
A function f(x) containing arbitrary constants such as a, b are called the general solutions of the differential equation. Also if the function f(x) does not contain an arbitrary constant, or contains some value assigned to the arbitrary constant, then they are called the particular solutions of the differential equation.
Differential Equation: d^{2}y/dx^{2} + 3dy/dx + 1 = 0
General Solution:y = 3x + k
Particular Solution: y = 3x + 2, Y = 3x + 7
The solutions containing arbitrary values are called the general solutions of the differential equation, and represent the family of curves if observed in the coordinate system. Further the solution without arbitrary constants or the solution obtained from the general solution by giving values to the arbitrary constants represents a particular curve in the coordinate axes, and it can be referred as the particular solution of a differential equation.
Related Topics
The following topics will help in a better understanding of solutions of a differential equation.
Examples on Solutions of A Differential Equation

Example 1: Find if the equation y = e^{2x }is a solution of a differential equation d^{2}y/dx^{2} + 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:
d^{2}y/dx^{2} = 4e^{2x}
Applying this in the differential equation to check if it satisfies the given expression.
The given differential equation is:
d^{2}y/dx^{2} + dy/dx 2y = 0
4e^{2x}  2e^{2x}  2e^{2x} = 0.
Therefore, the equation y = e^{2x }is a solution of a differential equation d^{2}y/dx^{2} + dy/dx 2y = 0.

Example 2: Verify if the function y = acosx + bsinx is a 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.
FAQs on Solutions of A Differential Equation
What Are The Solutions Of A Differential Equation?
The solution of a differential equation d^{n}y/dx^{n} + y =0 is an equation of a curve of the form y = f(x) which satisfies the differential equation. The differential equation has two types of solutions, general solution and a particular solution. The solution containing arbitrary constants is called a general solution and a solution without any arbitrary constants is called a particular solution of a differential equation.
How To Find the Solutions Of A Differential Equation?
The simplest method of finding the solutions of a differential equation is to segregate the variable and to integrate the functions distinctly to obtain the general solution of the differential equation. As an example the differential equation dy/dx = 2x/(3y 1) is segregated such that the expression of x and the derivative of x is one on side of the equals to sign, and the expression of y and the derivative on y on the other side.(3y  1).dy = 2x.dx. Further separate integration is performed to obtain the general solution.
What Are the Methods To Find Solutions Of Differential Equation?
The different methods to find the solutions of a differential equation is based on the type of the differential equation. The different types of differential equations are variable separable differential equation, homogeneous differential equations, nonhomogeneous differential equations, and linear differential equations.
What Is the Difference Between Particular Solution and General Solution Of A Differential Equation?
The difference between the general solution and particular solution is the presence of arbitrary constants. The solutions containing arbitrary values such as a, b, are called the general solutions of the differential equations. And the solution without arbitrary constants or the solution obtained from the general solution by giving values to the arbitrary constants is called a particular solution of a differential equation.
What Are the Arbitrary Terms In The Solution Of A Differential Equation?
The solution of a differential equation is an equation of a curve of the form y = f(x), and it has arbitrary constants a, b. And numeric values to these arbitrary constant transforms the general solution to a particular solution.
visual curriculum