Second order linear ODE: Difference between revisions
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= Solutions = | = Solutions = | ||
== Constant coefficient, homogeneous == | == Constant coefficient, homogeneous == | ||
Whatever derived usually also works for variable coefficients. | |||
These are the simplest kind. They have the general form | These are the simplest kind. They have the general form | ||
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Depending on the constants, it will give us anywhere from zero to two solutions: <math>y_1</math> and <math>y_2</math> | Depending on the constants, it will give us anywhere from zero to two solutions: <math>y_1</math> and <math>y_2</math> | ||
== Fundamental set of solutions == | |||
Given that two linearly independent solutions are given, the general solution is of the following form | Given that two linearly independent solutions are given, the general solution is of the following form | ||
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</math> | </math> | ||
The independence of the solutions can be checked using the [[Wronskian]]. | The independence of the solutions can be checked using the [[Wronskian]]. The two solutions are called a '''fundamental set''' of solutions. |
Revision as of 22:53, 21 May 2024
Second order linear ODEs are in the following form:
Important types of second order linear ODEs include
- Homogeneous
- Constant coefficients (where p and q are constants)
Initial value problem
There are two arbitrary constants in the solution of a second order linear ODE, so we need two initial conditions.
Solutions
Constant coefficient, homogeneous
Whatever derived usually also works for variable coefficients.
These are the simplest kind. They have the general form
We can guess the form of the solution to be exponential, since exponential functions has the property of being the same after many differentiation.
We substitute in the guess and obtain the characteristic equation
Depending on the constants, it will give us anywhere from zero to two solutions: and
Fundamental set of solutions
Given that two linearly independent solutions are given, the general solution is of the following form
The independence of the solutions can be checked using the Wronskian. The two solutions are called a fundamental set of solutions.