Suppose that the general solution to a differential equation

Suppose that the general solution to a differential equation is x(t)+ c_1[1 3] e^alphat + c_2[1 minus 1]e^minust for some constants alpha, c_1, and c_2. Sketch the phase portrait arising from the solution above in the three cases alpha = minus1, 0, 1 and describe the behavior of solutions as t rightarrow infin in each case.

Solution

Here, the Differential Equation can be written as

x(t) = A e^t + B e^-t

Now, the case1: when = -1, x(t) = A e^-1 + Be-¹ = (A + B)/e^t

When t tends  , x tends to 0

Case 2: When = 0 x(t) = A e⁰ + Be⁰ = (A + B)/e⁰ = A + B

When t tends  , x tends to A+B

Case 3: When = 1 x(t) = A e¹ + Be-¹ = A e¹ + B1/e^t

When t tends  , x tends to .