This page discusses general and particular solutions of differential equations.
Recommended Books on Amazon (affiliate links)  

When we solve differential equations, we will usually use integration. As you remember from integration, if you don't have initial conditions, you end up with a generic, unknown constant in your answer as a placeholder indicating that there are actually infinite solutions. The same concept appears when solving differential equations.
General vs Particular Solution
What is the difference between the general solution and the particular solution?
Short Answer   The terminology relates to whether or not we have an unknown constant in our final answer. For the general solution, an unknown constant IS part of the solution indicating an infinite number of solutions. In the particular solution, there is no unknown constant and the solution we have is the one and only solution.
Long Answer   When you solve a differential equation, you use integration, which introduces an unknown constant.
In the general solution, the unknown constant remains and you do not have enough information to be able to determine what that constant is. Consequently, you have an infinite number of solutions.
In the particular solution, you start out by solving for the general solution but then you are given initial conditions which you use to determine the value of the constant. These initial conditions are actually points that the solution to the differential equation pass through. In the end, you have only one solution without any unknown constants.
general solution 
particular solution  

infinite number of solutions 
only one solution  
contains unknown constant(s) 
does NOT contain any unknown constants  
no (or too few) initial conditions given 
initial conditions given; used to solve for constants 
To find the general solution, just solve the differential equation and leave any unknown constants in your final answer.
To find the particular solution, find the general solution first, then plug in the initial conditions and solve for the constants.
Notes
1. When determining the particular solution, you will be given the same number of initial conditions as the order of the differential equation. Depending on how the initial conditions are given, you may need to stop after each integration and solve for the individual constants or you may need to wait until you are completely done and solve for all the constants at once. You will get a feel for this as you work practice problems.
2. For a general solution, if no initial conditions are given or fewer than the order of the differential equation are given, we cannot determine all of the unknown constants, since each integration introduces another constant.
3. When you were first learning integration, you probably ran across initial value problems. These were actually differential equations where you were asked to find the particular solution.
Practice
Verify that \( y = x^3  c/x \) is a solution to \( 2xy' + 2y = 8x^3 \) and find the value of \(c\) for which the solution satisfies the initial condition \(y(2) = 6\).
Problem Statement
Verify that \( y = x^3  c/x \) is a solution to \( 2xy' + 2y = 8x^3 \) and find the value of \(c\) for which the solution satisfies the initial condition \(y(2) = 6\).
Solution
video by MIP4U 

Log in to rate this practice problem and to see it's current rating. 

Verify that \( y=c_1 \sin(2x) + c_2 \cos(2x) + 1 \) is the general solution to \( y'' + 4y = 8 \) and solve for \(c_1\) and \(c_2\) if \(y(0) = 5\) and \(y'(0) = 1\).
Problem Statement
Verify that \( y=c_1 \sin(2x) + c_2 \cos(2x) + 1 \) is the general solution to \( y'' + 4y = 8 \) and solve for \(c_1\) and \(c_2\) if \(y(0) = 5\) and \(y'(0) = 1\).
Solution
video by blackpenredpen 

Log in to rate this practice problem and to see it's current rating. 

Verify that \( y=Ce^x  3x  3 \) is the general solution to \( dy/dx = 3x + y \) and solve for \(C\) if \(y(0) = 1\).
Problem Statement
Verify that \( y=Ce^x  3x  3 \) is the general solution to \( dy/dx = 3x + y \) and solve for \(C\) if \(y(0) = 1\).
Solution
video by blackpenredpen 

Log in to rate this practice problem and to see it's current rating. 

Verify that \( y=x^2 + c/x^2 \) is the general solution to \( xy' + 2y = 4x^2 \) and solve for \(c\) if \( y(5) = 8 \).
Problem Statement
Verify that \( y=x^2 + c/x^2 \) is the general solution to \( xy' + 2y = 4x^2 \) and solve for \(c\) if \( y(5) = 8 \).
Solution
video by MIP4U 

Log in to rate this practice problem and to see it's current rating. 

The general solution of a differential equation is \( y = ae^{4t} + be^{6t} \) where \(a\) and \(b\) are constants. Use the initial conditions \(y(0) = 2\) and \(y'(0) = 5 \) to find the particular solution.
Problem Statement
The general solution of a differential equation is \( y = ae^{4t} + be^{6t} \) where \(a\) and \(b\) are constants. Use the initial conditions \(y(0) = 2\) and \(y'(0) = 5 \) to find the particular solution.
Solution
video by MIP4U 

Log in to rate this practice problem and to see it's current rating. 

Verify that \( y = ce^{2x} + e^{x} \) is a solution to the differential equation \( y' + 2y = e^{x} \) and determine the value of \(c\) for which the solution satisfies the initial condition \( y(4) = 4 \).
Problem Statement
Verify that \( y = ce^{2x} + e^{x} \) is a solution to the differential equation \( y' + 2y = e^{x} \) and determine the value of \(c\) for which the solution satisfies the initial condition \( y(4) = 4 \).
Solution
video by MIP4U 

Log in to rate this practice problem and to see it's current rating. 

Really UNDERSTAND Differential Equations
Log in to rate this page and to see it's current rating.
To bookmark this page and practice problems, log in to your account or set up a free account.
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
 
Join Amazon Prime  Watch Thousands of Movies & TV Shows Anytime  Start Free Trial Now 

I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me. 

Support 17Calculus on Patreon 

