Solution of Differential Equation

General  and Particular Solution of  Differential Equation

The solution of a differential equation is the relationship between the variables included which satisfies the differential equation. There are two types of solutions of differential equations namely, the general solution of differential equations and the particular solution of the differential equations. The general and the particular solutions of differential equations make use of some steps of integration to solve the equations. There are 5 methods to solve the differential equation.

These 5 Methods Are:

  1. Solution by inspection

  2. Variable separable

  3. Homogeneous

  4. Linear differential equation

  5. General

In this article we will learn the general and the particular solution of differential equation, single solution of differential equation, how to find the particular solution and general solution of the differential equation with the help of  differential equation practice problems with solution etc.


What is the Differential Equation?

A differential equation is an equation which includes one or more terms and also includes the derivatives of one variable (i.e., dependent variable) in terms of the other variable (i.e., independent variable)

dt/dz = f(z)

Here “z” is an independent variable and “t” is a dependent variable

For example, dt/dz = 5z


General Solution of  a Differential Equation

A General Solution of nth order differential equation is defined as the solution that includes n important arbitrary constants.

It is necessary for us to introduce an arbitrary constant as soon as integration is performed if we solve a first order differential equation by a variable method. Hence, you can see after simplifying that the solution of the differential equation of first order includes an important arbitrary constant.

Similarly, the general solution of a second order differential equation will include important arbitrary constants and so on. Geometrically,the general solution represents an n-parameter family of curves. For example, the general solution of the differential equation dy/dx = 8x² which is found to be y = x³ + C,  where c is considered as  an arbitrary constant, represents a one-parameter family of curves as shown in the figure given below.

(Image to be added soon)


Particular Solution of  a Differential Equation

The particular solution of a differential equation is a solution which we get from the general solution by giving particular values to an arbitrary solution. The conditions for computing the values of arbitrary constants can be given to us in the form of an initial-value problem or Boundary Conditions depending on the questions.


Singular Solution of a Differential Equation

The singular solution of a differential equation is a special kind of particular solution of a differential equation but it cannot be derived from the general solution of a differential equation by assigning the values of the random constant.


Differential Equation Practice Problems

Here, you can see some of the differential equation practice problems.

  1. Find the general solution of the following differential equation

dt/dx = (1 + x²) ( 1+ t²)

Solution:

The given differential equation is dt/dx = (1 + x²) ( 1+ t²)

dt( 1+ t²) = (1 + x²)dx

By integrating both sides of the above equation, we get

∫dt/( 1+ t²) = ∫(1 + x²)dx

tan-1 t = ∫dx ∫dx²

tan-1 t  = x + x³/x + C

The above equation is the required general solution of the differential equation.


  1. Find the general solution of the differential equation given below

dt/dz = ez + t  

Solution: 

We have,

dt/dz = ez + t  

Using the law of exponent, we get

dt/dz = ez + et

By separating variables by variable separable procedure, we get

e-t dt = ez dz

Now taking integration of both the side, we get

∫e-t dt = ∫ez dz

On integrating, we get

  • -e-t = ez + C

  • ez + e-t  = - C  Or  ez + e-t =  c


Differential Equation Practice Problems With Solutions

Here, you can see some of the differential equation practice problems with solutions

  1. Find the particular solution of a differential equation which satisfies the below condition

dy/dx = 3x2 – 4 ; y(0) = 4

Solution:

We will first find the general solution of a differential equation. To do this, we will integrate both sides to find y 

dy/dx = 3x2 – 1  

y = ∫(3x2 – 1) dx

y = x3 –x + 4

This is our general solution, to find the particular solution of a differential equation, we will apply the initial condition given to us ( y= 4 and x = 0) and solve for C:

y = x3 – x + c

Now, we apply our initial conditions (x = 0, y = 4) and solve for C, which will give us our particular solution:

4 = (0)3 – 0 + C

Now, we will solve for C

4 = C

y = x3 –x + 4

Hence, the particular solution of a differential equation is x3 –x + 4


  1. Find the particular solution of a differential equation which satisfies the below condition

dy/dx = 1/x2 ; y(1) = 4

Solution:

We will first find the general solution of a differential equation. To do this, we will integrate both sides to find y 

dy/dx = 1/x2 ;

y(1) = 4

This is our general solution, to find the particular solution of a differential equation, we will apply the initial condition given to us ( y= 4 and x = 1) and solve for C:

y = -1/x+ c 

y= ∫(1/x2) dx 

y= ∫(1/x2) dx

y = x-1 / -1 + C

Now, we apply our initial conditions (x = 1, y = 4) and solve for C, which will give us our particular solution:

4 = -1/1+C

Now, we will solve for C

4 = -1 + C

5 = C

y = -1/x + 5

Hence, the particular solution of a differential equation is y = -1/x + 5


Quiz Time

  1. The differential solution is considered to be ordinary when it has

  1. One dependent variable

  2. More than one dependent variable

  3. One independent variable

  4. More than one independent variable.

  1. A general solution of 3 degree order differential equation includes _________ constants

  1. 1

  2. 2

  3. 3

  4. 4

  1. A general solution of 4 degree order differential equation includes _________ constants

  1. 1

  2. 2

  3. 3

  4. 4

  1. Find the particular solution of the differential equation

dy/dx = 4x-2

When y= 5 and x= 3

  1. 2x² - 2x + 7

  2. 5x² - 3x + 7

  3. 9x² - 1x + 5

  4. 6x² - 3x +4

FAQ (Frequently Asked Questions)

1. What Are The Different Applications of Differential Equations?

Here are some of the different applications of differential equations in real-time:

  • Differential equations detail different exponential growths and declines

  • Through the differential equation, we can know the rate of change in investment return over a period of time.

  • They are used in the field of health care for modeling cancer growth or the spread of various diseases in the human body.

  •  Electricity movenemt can also be reported with the help of the differential; equation.

  • Differential equations help economists to discover feasible investment strategies.

  • The motion of waves or a pendulum can also be reported through these quadratic equations

  • It is also used to figure out the motion of waves in physics.

2. Explain The Solution of Differential Equations.

The general solution of the differential equation is the correlation between the variables x and y which is received after removing the derivatives (i.e. integration) where the relation includes arbitrary constants to represent the order of an equation. The solution of the first- order differential equation includes one arbitrary whereas the second- order differential equation includes two arbitrary constants. If specific values are given to arbitrary constants, the general solution of the differential equation is received. In order to resolve the first-order differential equation of first degree, some standard forms are here to obtain the general solution as follows.

  • Linear equation

  • Homogeneous equation’

  • Non-homogeneous differential equation

  • Exact differential equation

  • Variable desperate method