Question

Nonhomogeneous linear system. Consider the nonhomogeneous system

x′=P(t)x+g(t),g(t)=
(0)
(0)
(t)
⎛where the 3×3 matrix P(t) is continuous for −[infinity]X(t)=[x₁​(t),x₂​(t),x₃​(t)]=
[1 eᵗ 0]
[-1 eᵗ(1+t) eᵗ]
[-1 eᵗt eᵗ]
Use the method of variation of parameters to find the solution of the original nonhomogeneous system subject to the initial condition x1​(0)=0,x₂​(0)=0, and x₃​(0)=0. (No need to find P(t). )

Answer 1

The method of variation of parameters can be used to find the solution of the nonhomogeneous system. It involves finding the general solution of the corresponding homogeneous system and using a variation of parameters formula to find a particular solution. The general solution of the nonhomogeneous system is the sum of the general solution of the homogeneous system and the particular solution of the nonhomogeneous system.

Step-by-step explanation:

The method of variation of parameters can be used to find the solution of the original nonhomogeneous system. To find the solution, we first need to find the general solution of the corresponding homogeneous system x' = P(t)x.

Then, we can use the variation of parameters formula to find a particular solution of the nonhomogeneous system. Finally, the general solution of the nonhomogeneous system can be expressed as the sum of the general solution of the homogeneous system and the particular solution of the nonhomogeneous system.