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Solving linear second order differential equations with constant coefficients

Describes the process of solving these equations and gives some examples.
by

David Butler

on 2 December 2013

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Transcript of Solving linear second order differential equations with constant coefficients

Original equation
1. Try something like r(x) but with unknown coefficients:
2. From y, find y' and y'':
4. Find values of unknown coefficients:
5. Put the values in:
eg: If
then try
then try
eg: If
then try
eg: If
then try
eg: If
Equate coefficients of powers of x
Sub in values of x
OR
Particular solution to original equation
number
number
number
function of x
General solution to homogeneous equation
1. Write down characteristic equation:
3. Use these values to make two solutions:
4. Use these two solutions to make general solution:
If
then use
then use
If
If
then use
2. Solve characteristic equation:
&
&
&
Solving linear second-order differential equations with constant coefficients
3. Sub into original equation:
General solution to original equation
1. Add the general solution for the homogeneous equation
to the particular solution for the original equation:
By David Butler (c) The University of Adelaide August 2010
Note: The theory says that if I have two solutions that aren't multiples of each other, then all the rest of the solutions must be linear combinations of these two.
That's why we just make a linear combination of them (with arbitrary coefficients) to get the general solution.
Note: If this doesn't work, try including a term like "Bxe^(-4x)" too.
Note: If this doesn't work, try including a term like "Dx^3" too.
Note: If this doesn't work, try including a term like "Bx" as well.
Note: If this doesn't work, try including terms like
"Cx cos(2x)" and "Dx sin(2x)" too.
1. If there are initial conditions, such as y(0)=0 and y'(0) =5
sub them in now to find C and D.
Example
Particular solution to original equation
satisfying conditions

2. Put the values of C and D in to get the final solution.
Note: Don't try including them earlier than this - you must wait until you've found the general solution.
Note: If there are initial conditions, this solution is not likely to satisfy them.
We are going to use this solution to find ALL the other solutions.
After that we can find another particular solution that DOES satisfy the conditions.
Example
Full transcript