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# Solve an IVP ODE using the Method of Variation of Parameters

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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

https://brainmass.com/math/numerical-analysis/solve-ivp-ode-method-variation-parameters-6525

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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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