MATH 1014.03MW **Exam 4** Monday, March, 27 2000

NAME:

Student Number: No.Marks

**Instructions**: This exam contains 4 questions and has a total
of 100 marks. Show all of your work. Time: 35 minutes.

(20 Marks) Solve the following equation

Solution (Method 1) Rewrite the equation as

Integrating the above equation from 0 to *t*, we have

and

This implies .

(Method 2) The general solution of the equation is

Since s(0)=1000, we have *C*=1000. Hence, the required solution is

(20 Marks)Find the general solution of the equation

Solution Rewrite the equation as

The integrating factor

The general solution

(25 Marks)Find the general solution of the following
equation.

Solution (1) Find the general solution of *y*''+6*y*'+9*y*=0.

The auxiliary equation is . Solving the equation, we have
*r*=-3. The general solution

(2) [Note that and are particular solutions of
*y*''+6*y*+9*y*=0]

Let . Then

This implies 2*C*=1 and *C*=1/2. Hence,

(3) The required general solution is

(25 Marks)Let and
. Prove that

Solution Taking the partial derivatives of , we obtain

This implies and . Since

and

we obtain

Mon Mar 27 10:04:47 EST 2000