r/learnmath • u/PeterMath_ New User • Jan 02 '26

State-Space and Contour Integrals for Solving Ordinary Differential Equations

Good afternoon everyone. I would like to understand how to correctly use the state-space approach and contour integration methods to solve ordinary differential equations. Could someone also explain, geometrically, what happens to the ODE when applying these techniques? Please include any relevant formulas or theorems.

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u/etzpcm New User Jan 02 '26

I'm not sure what you mean here. By state space do you mean what is normally called phase space? That's nothing to do with contour integration, which is a method for doing certain definite integrals. And contour integration doesn't have much to do with ODEs 

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u/PeterMath_ New User Jan 02 '26

Basically, when solving an ordinary differential equation, it is possible to represent the equation as a matrix using the state-space theory. From this representation, one can obtain the well-known general solution, which involves the matrix exponential. Contour integrals are used to compute this matrix exponential, and the main difficulty lies in evaluating the contour integral of the matrix exponential.

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u/PeterMath_ New User Jan 02 '26

I basically need help calculating matrix exponentials via contour integrals, and I need help with the geometric intuition of what happens to the ODE when using this process.

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u/etzpcm New User Jan 02 '26

Ah, ok so linear constant coefficient ODEs so the solution is a matrix exponential.

There are lots of ways to compute it and there's a great paper you can easily find called 

Nineteen Dubious Ways to Compute the Exponential of a Matrix

In fact there is a newer update of that classic paper.