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Solve an IVP ODE using the Method of Variation of Parameters

Please see the attached file for the fully formatted problems.

Solve an IVP ODE using the method of variation of parameters

Find the solution of the system X'
using the method of variation of parameters

2 0 0 cos(t)
X' = -1 0 -1 X + sin(t)
1 1 2 e^-t

that satisfies the intial condition

( 0 )
X(0) = 1
-1

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Solution.

From what you provided, we can rewrite the differential equations as follows. Let X=(x(t),y(t),z(t))', or X=(x,y,z)' in short, then we have
x'=2x+cost (1)
y'=-x-z+sint (2)
z'=x+y+2z+e(-t) (3)
By (1), it is a first order ordinary linear differential equation. You can use a formula in your text book and get the following solution.
x(t)=Ae^(2t)+1/5*sint-2/5*cost.
Using the initial condition x(0)=0, we can determine the constant A. By a simple calculation we get
A=2/5
So ...

Solution Summary

An Initial Value Problem - Ordinary Differential Equation is solved using the method of variation of parameters.

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