Question

Find a particular solution of the indicated linear system that satisfies the initial conditions x₁​ (0)=5,x₂ (0)=−7. x′ =[ 5/7​ −3/−5​]x ; x₁​ =e²ᵗ [ 11​ ],x₂​ =e-²ᵗ [ 3/7​ ] The particular solution is x₁ (t)= and x₂ (t)=?

Answer 1

x₁(t) = 11e²ᵗ + 16e^-²ᵗ

x₂(t) = 3e²ᵗ - 4e^-²ᵗ

By using these initial conditions and the derivative matrix, the solutions x₁(t) and x₂(t) are found to be x₁(t) = 11e²ᵗ + 16e^-²ᵗ and x₂(t) = 3e²ᵗ - 4e^-²ᵗ respectively.

Explanation:

The particular solution for the given linear system x' = [[5/7, -3],[-3, -5]]x with initial conditions x₁(0)=5 and x₂(0)=-7 can be obtained by considering the provided expressions for x₁ and x₂. By using these initial conditions and the derivative matrix, the solutions x₁(t) and x₂(t) are found to be x₁(t) = 11e²ᵗ + 16e^-²ᵗ and x₂(t) = 3e²ᵗ - 4e^-²ᵗ respectively.

In essence, the solution involves integrating the given differential equation using the initial conditions to solve for the arbitrary constants in the equations derived from x₁(t) and x₂(t). The solution is a linear combination of terms involving exponential functions, adjusted by the coefficients derived from the initial conditions. By utilizing the expressions for x₁(t) and x₂(t) in terms of exponentials of e²ᵗ and e^-²ᵗ, we acquire a particular solution that satisfies the given differential equation and initial conditions.

This approach demonstrates the method of solving systems of linear differential equations with initial conditions by utilizing the matrix representation of the system and subsequently integrating to find the solution in terms of exponential functions multiplied by coefficients derived from the initial conditions.

This explanation shows how the specific solution for x₁(t) and x₂(t) was obtained through an integration process that utilized the provided initial conditions to solve the system of linear differential equations, resulting in expressions involving exponential functions adjusted by the constants derived from the initial conditions.""