General existence theorem: Difference between revisions
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[[Category:Differential Equations]] | |||
Every [[Second order linear ODE|Second order linear homogeneous ODE]] with continuous coefficients has a fundamental set of solutions. | Every [[Second order linear ODE|Second order linear homogeneous ODE]] with continuous coefficients has a fundamental set of solutions. | ||
This is proven by handpicking | This is proven by handpicking two sets of initial conditions such that the [[Wronskian]] is 1 for two specific solutions to the homogeneous. | ||
Now that we have the two, every other IVP is solved (pretty easily because variables are nice). | |||
This also works for [[systems of ODEs]]. | |||
= General solution = | |||
A strategy for solving for the general solution is then to take a particular set of fundamental solutions for a IVP. We can then solve other IVPs with the fundamental set for c (the constant coeffiicients of the linear combination). Then, by the [[existence-uniqueness]] theorem, the linear combination has to be the only solution. | |||
Latest revision as of 14:58, 10 June 2024
Every Second order linear homogeneous ODE with continuous coefficients has a fundamental set of solutions.
This is proven by handpicking two sets of initial conditions such that the Wronskian is 1 for two specific solutions to the homogeneous.
Now that we have the two, every other IVP is solved (pretty easily because variables are nice).
This also works for systems of ODEs.
General solution
A strategy for solving for the general solution is then to take a particular set of fundamental solutions for a IVP. We can then solve other IVPs with the fundamental set for c (the constant coeffiicients of the linear combination). Then, by the existence-uniqueness theorem, the linear combination has to be the only solution.
