Question

Consider the system of differential equations [5 31 2, 3 5 Verify that (0) = 13 (t) = C1e2t * 1] +ox" [] is a solution to the system of differential equations for any choice of the constants C1 and C2. Find values of C1 and C2 that solve the given initial value problem. (According to the uniqueness theorem, you have found the unique solution of T' = P7, 7(0) = 7o). (t) = = 1)--[-1]+(O)--[1] help (numbers)

Answer 1

To verify the given solution, substitute the values of x'(t) and x(t) into the differential equations and check if they satisfy the equations. The values of C1 and C2 that solve the initial value problem are C1 = (33/37)e^-2t and C2 = (1/2)e^-2t.

Step-by-step explanation:

To verify that the given solution is indeed a solution to the system of differential equations, we need to substitute the values of x'(t) and x(t) into the differential equations and check if they satisfy the equations. Starting with the first equation:

5x'(t) + 31x(t) = 2(1 - x(t))

Substituting the given values of x'(t) and x(t):

5(2C1e^2t) + 31(C1e^2t - 1) = 2(1 - C1e^2t)

Simplifying:

10C1e^2t + 31C1e^2t - 31 = 2 - 2C1e^2t

39C1e^2t - 2C1e^2t = 33

37C1e^2t = 33

C1e^2t = 33/37

C1 = (33/37)e^-2t

Now, let's check the second equation:

3x'(t) + 5x(t) = 1 - x(t)

Substituting the given values of x'(t) and x(t):

3(2C1e^2t) + 5(C1e^2t - 1) = 1 - C1e^2t

Simplifying:

6C1e^2t + 5C1e^2t - 5 = 1 - C1e^2t

11C1e^2t + C1e^2t = 6

12C1e^2t = 6

C1e^2t = 1/2

C1 = (1/2)e^-2t

Therefore, the values of C1 and C2 that solve the initial value problem are C1 = (33/37)e^-2t and C2 = (1/2)e^-2t.