Question

Solve the initial value problem for the following system of first order differential equations

dt/dx₁ =−2x₁+2x₂+f(t)
dt/dx₂ =x₁−3x₂+x₁0)=x₂ (0)=0​
where f(t) is a given function.

Answer 1

To solve the given system of first order differential equations, we can use the method of variation of parameters. We need to find x₁ and x₂ such that dx₁/dt = -2x₁ + 2x₂ + f(t) and dx₂/dt = x₁ - 3x₂. By solving these equations using the method of variation of parameters, we can find the general solution of the system in terms of constants C₁ and C₂ along with functions u₁(t) and u₂(t).

Step-by-step explanation:

To solve the given system of first order differential equations, we can use the method of variation of parameters. Let's denote the solution of the system as x₁ and x₂. We need to find x₁ and x₂ such that:

dx₁/dt = -2x₁ + 2x₂ + f(t) ......(1)

dx₂/dt = x₁ - 3x₂ ......(2)

To find x₁ and x₂, we start by finding the general solution of the homogeneous system (when f(t) = 0). Let's assume x₁ = C₁e^(λt) and x₂ = C₂e^(λt).

By substituting these values in equations (1) and (2), we get:

λC₁e^(λt) = -2C₁e^(λt) + 2C₂e^(λt)

λC₂e^(λt) = C₁e^(λt) - 3C₂e^(λt)

Dividing both equations by e^(λt), we get:

λC₁ = -2C₁ + 2C₂

λC₂ = C₁ - 3C₂

This gives us a system of linear equations which we can solve to obtain the values of C₁ and C₂. Once we have the values of C₁ and C₂, we can substitute them back into the equations x₁ = C₁e^(λt) and x₂ = C₂e^(λt).

To find the particular solution when f(t) is not equal to 0, we can use the method of variation of parameters. We assume x₁ = u₁(t)x₁(t) and x₂ = u₂(t)x₂(t), where u₁(t) and u₂(t) are unknown functions.

By substituting these values into equations (1) and (2), we get:

u₁(t)dx₁/dt + x₁(t)du₁/dt = -2x₁ + 2x₂ + f(t)

u₂(t)dx₂/dt + x₂(t)du₂/dt = x₁ - 3x₂

Simplifying and rearranging the equations, we get:

dx₁/dt = (-2x₁ + 2x₂ + f(t))/u₁(t)

dx₂/dt = (x₁ - 3x₂)/u₂(t)

From equations (1) and (2), we have dx₁/dt = -2x₁ + 2x₂ + f(t) and dx₂/dt = x₁ - 3x₂. Solving these equations, we can find the values of u₁(t) and u₂(t).

Therefore, the general solution of the given system of first order differential equations is x₁ = C₁e^(λt) + u₁(t)x₁(t) and x₂ = C₂e^(λt) + u₂(t)x₂(t), where C₁ and C₂ are constants and u₁(t) and u₂(t) can be obtained by solving the equations involving dx₁/dt and dx₂/dt respectively.