# Solving Initial Value Problem The implemented Maple packege is based on the convertion of the given system into a canonical form using the shuffle algorithm which produces another simple equivalent system, and the canonical system can be solved easily.The Maple implementation includes computing the canonical system and the exact solution of a given IVP.

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.

We get $(s^2\mathcal\ - 2s - 1) (s\mathcal\ - 2) - 2\mathcal\ = 4/s.$ Next, combine like terms to get $(s^2 s - 2)\mathcal\ = 4/s 2s 3. However, we recall a symbolic algorithm to compute the exact solution of a given system of DAEs (See  for further details of the algorithm). In this paper, we discuss the Maple package of the symbolic algorithm that computes the exact solution.$ Notice that the coefficient in front of $$\mathcal$$ is the characteristic equation of the differential equation. Putting under a common denominator, dividing and factoring we get $\mathcal\ = \dfrac .$ To find $$y$$, we need to take the Inverse Laplace Transform of the right hand side.

Unfortunately, finding a function $$y$$ such that the right hand side is the Laplace transform of $$y$$ is not an easy task.

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.

## Comments Solving Initial Value Problem

• ###### Choose an ODE Solver - MATLAB & Simulink
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In an initial value problem, the ODE is solved by starting from an initial state. Using the initial condition, y 0, as well as a period of time over which the answer is to be obtained, t 0, t f, the solution is obtained iteratively. At each step the solver applies a particular algorithm to the results of previous steps.…

• ###### SCF11 text Laplace Solving Initial Value Problems
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Laplace Solving Initial Value Problems OCW 18.03SC Example 4. Find the unit impulse response for the system pDx = f, where pD = D2 +2D +2I and we consider f to be the input.…

• ###### Antiderivatives and Initial Value Problems - math.dartmouth.edu
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Initial value problems Find a solution to the di↵erential equation d dx y = x2 +1which also satisﬁes y2 = 8/3. general solution y = 1 3 x 3 +x +C particular solution y = 1 3 x 3 +x 2 Each color corresponds to a choice of C. Red cuve is the particular solution.…

• ###### Solving Boundary Value Problems for Ordinary Di erential Equations in.
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This is an initial value problem IVP. However, in many applications a solution is determined in a more complicated way. A boundary value problem BVP speci es values or equations for solution components at more than one x. Unlike IVPs, a boundary value problem may not have a solution, or may have a nite number, or may have in nitely many.…

• ###### Solve the initial value problem -
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Let's look at solving this with a slightly different method. The matrix equation is equivalent to the pair of equations $\displaystyle y_1'= 2y_1+ y_2+ e^x$ and $\displaystyle y_2'= -y_1+ 2y_2$. Differentiate the first equation again to get $\displaystyle y_1''= 2y_1'+ y_2'+ e^x$.…

• ###### Chapter 5 The Initial Value Problem for Ordinary Differential Equations
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The initial value problem may fail to have a unique solution over any time interval if this initialvalue is imposed. Example5.4. Considerthe initialvalue problem u0.t/D p u.t/ withinitialcondition u.0/D 0 The function f.u/ D p u is not Lipschitz continuous near u D 0 since f0.u/ D 1=.2 p u/!1as u ! 0.…

• ###### Elementary Calculus Solve the Initial Value Problem
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Solve the Initial Value Problem. -2y = 0 when y = 0. Thus the constant yt = 0 is a particular solution. where C = eB if y 0, and C = -eB if y 0. General solution yt = Ce-2t, where C is any constant C can be 0 from Step 1. Substitute 1 for t and - 5 for y, and solve for C. -5 = Ce-21, -5 = Ce-2, C = -5e2.…

• ###### WHAT IS A DIFFERENTIAL EQUATION?
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Value problem. INITIAL VALUE PROBLEM. The problem of finding a function y of x when we know its derivative and its value y. 0. at a particular point x. 0. is called an initial value problem. This problem can be solved in two steps. 1. 2. Using the initial data, plug it into the general solution and solve for c. EXAMPLE 1 Solve the initial.…

• ###### Solve an Initial Value Problem for the Wave Equation New in. - Wolfram
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Solve the initial value problem with a sum of exponential functions as initial data.…