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Consider the following second order liner singular system $K x''( t ) = A x'( t )$,where $K$ is a singular matrix,$A$ is any matrix,$x(0)=\left<2, 1\right>,x'(0)=\left<0, 1\right>$. how to solve the above system of differential equations?

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If $K$ is singular, the system is no longer an ODE but a DAE, a differential-algebraic system. Since it is linear, you can solve it with the Kronecker normal form.

Or use any of the index theories. If $K$ and $A$ are sparse, you can use Pantelides/Pryce theory of the Taylor index.

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