Solution of a Differential Equation

A differential equation involves derivatives of an unknown function, and its solution is usually a function rather than a single number.

This topic explains the meaning of a solution of a differential equation and distinguishes between a general solution and a particular solution.

Any relation between independent and dependent variables which does not involve derivatives, such that this relation and the derivatives obtained from it satisfy the given differential equation, is called a solution of the differential equation.

A solution of a differential equation in which the number of arbitrary constants equals the order of the differential equation is called the general solution of the differential equation.

A solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution.

Verification of a particular function

Verify that \[y = e^{-3x}\] is a solution of \[\frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 0\].

Step 1: Differentiate once

\[\frac{dy}{dx} = -3e^{-3x}\]

Step 2: Differentiate again

\[\frac{d^2y}{dx^2} = 9e^{-3x}\]

Step 3: Substitute into the differential equation

\[\frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 9e^{-3x} - 3e^{-3x} - 6e^{-3x} = 0\]

\[9e^{-3x} - 9e^{-3x}\] = 0

Conclusion: Hence, \[y = e^{-3x}\] is a solution of the given differential equation.

Show that \[ y = a\sin(x+b), \qquad a, b \in R \] is a general solution of \[ \frac{d^{2}y}{dx^{2}} + y = 0. \] 

Also, obtain a particular solution when \[ a = 2, \qquad b = \frac{\pi}{4}. \]

Solution:

Given, \[ y = a\sin(x+b) \]

Differentiating, \[ \frac{dy}{dx} = a\cos(x+b) \]

Again differentiating, \[ \frac{d^{2}y}{dx^{2}} = -a\sin(x+b) \]

Therefore, \[ \frac{d^{2}y}{dx^{2}} + y = -a\sin(x+b) + a\sin(x+b) = 0 \]

Hence, \[ \boxed{\,y = a\sin(x+b)\,} \] is a general solution of \[ \frac{d^{2}y}{dx^{2}} + y = 0. \]

Now, putting

\[ a = 2, \qquad b = \frac{\pi}{4}, \] we get

\[ \boxed{\,y = 2\sin\left(x + \frac{\pi}{4}\right)\,} \]

which is a particular solution of the differential equation.

• A differential equation contains derivatives of an unknown function.

• Its solution is generally a function, not a single number.

• The graph of the solution function is called the solution curve or integral curve.

• A general solution contains arbitrary constants.

• A particular solution is obtained by assigning fixed values to those constants.

• To verify a solution, substitute the function and its derivatives into the equation and check whether LHS = RHS.