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- Homework Statement
- How to solve the following matrix differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##, where ##A(t)## and ##B(t)## are ##n\times n## matrices and ##\left | \psi \right >## is a ##n##-vector.

- Relevant Equations
- None

Hello, there. I am trying to solve the differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##. However, ##A(t)## and ##B(t)## can not be simultaneous diagonalized. I do not know is there any method that can apprixmately solve the equation.

I suppose I could write the equation as ##\partial_t \left | \psi \right >=-B^{-1}(t) A(t)\left | \psi \right > ## with general solutions being ##\left | \psi \right >=\exp \left ( \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ## and ##\left | c\right > ## is a constant vector. Then a first-order approximation may be ##\left | \psi \right >=\left (I+ \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ##.

I am not familiar with matrix differential equations. Does this method have any restrictions or problems? Or there may be other better approximation solutions? Any references would be greatly appreciated.

Thanks!

I suppose I could write the equation as ##\partial_t \left | \psi \right >=-B^{-1}(t) A(t)\left | \psi \right > ## with general solutions being ##\left | \psi \right >=\exp \left ( \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ## and ##\left | c\right > ## is a constant vector. Then a first-order approximation may be ##\left | \psi \right >=\left (I+ \int_0^t -B^{-1}(t') A(t')dt'\right ) \left | c\right > ##.

I am not familiar with matrix differential equations. Does this method have any restrictions or problems? Or there may be other better approximation solutions? Any references would be greatly appreciated.

Thanks!