Question

Consider the following second-order differential equation: ?"(?)−?′(?)−6?(?)=?(?)

(1) Let ?(?)=−12e^t. Find the general solution to the above equation.

(2) Let ?(?)=−12.

a) Convert the above second-order differential equation into a system of first-order differential equations.

b) For your system of first-order differential equations in part a), find the characteristic equation, eigenvalues and their associated eigenvectors.

c) Find the equilibrium for your system of first-order differential equations. Draw a phase diagram to illustrate the stability property of the equilibrium.

Answer 1

(1)

Considering in the equation

gives .

Here auxiliary equation of the associated homogeneous equation is .

Calculating gives .

Hence, roots will be and .

Hence, CF of the above differential equation will be .

Now to determine PI we calculate

.

Hence, the required general solution will be= CF+PI=.