# Math 0250 Homework 8

• 2.6.1 Number 1
Since , we have

• 2.6.1 Number 3

• 2.6.1 Number 5
Use the identities

to get

• 2.6.1 Number 6
Look for a solution of the form X = K e2t. Substituting this in the equation X'=AX+F gives

Cancel the exponential and rearrange terms to get

and solve to get

so a particular solution is

• 2.6.1 Number 7
First complexify the equation. The forcing function is the real part of

so let's solve the system

and then take the real part of the result to get a solution to the original system. Look for a solution of the form Z=Keit. Substituting into the above system and canceling the exponentials gives

and rearranging gives

Solving for K gives

so

and a particular solution to the original system is

• 2.6.1 Number 10
First complexify the equation. The forcing function is the real part of

so let's solve the system

and then take the real part of the result to get a solution to the original system. Look for a solution of the form Z=Ke(-1+2i)t. Substituting into the above system and canceling the exponentials gives

and rearranging gives

Solving for K gives

so

and a particular solution to the original system is

• 2.6.2 Number 1
The solution has the form where V satisfies

Solving for V' gives

V can now be obtained by integration. You can use a computer, a table, or bare hands to evaluate the integral. A standard textbook trick is to integrate by parts twice. A more direct method is to observe that the two entries of V' are the real and imaginary parts of e(1+i)t, and

It follows that

Since we're only looking for one solution, we'll drop the constant of integration, to get

• 2.6.2 Number 2
The solution has the form where V satisfies

Solving for V' gives

It follows that

Since we're only looking for one solution, we'll drop the constant of integration, to get

Frank Beatrous
1998-11-03