# Solving Initial Value Problem

Solution We could solve this problem using the method of undetermined coefficients, however that would involve finding $$y_h$$, $$y_p$$, and the two constants.

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We have Proof To prove this theorem we just use the definition of the Laplace transform and integration by parts.

We will prove the theorem for the case where $$f$$' is continuous.

We begin by applying the Laplace transform to both sides.

By linearity of the Laplace transform, we have \[ \mathcal\ \mathcal\ - 2\mathcal\ = \mathcal\ . There is on in the textbook or you can find one online here.

The lemma gives the variation of parameters formula of an IVP for higher-order scalar linear differential equations over integro-differential algebras.

(Thota and Kumar [1, 2, 5, 7, 9]) Let obtained from W by replacing the i-th column by m-th unit vector.

Then, we could find values of parameters by substituting the initial conditions.

For example, the implemented method in Mathematica is based on decomposing the coefficient matrices, A and B, into nonsingular and nilpotent part.

Then generalized inverse for A and B is calculated, and the problem is reduced to solving a system of ODEs. In Matlab, the equation is also converted to system of ODEs by reducing the differential index and then we find the general solution with free parameters.

However, in the proposed algorithm, we compute the exact solution directly without free parameters.

## Comments Solving Initial Value Problem

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