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DEPARTMENT OF MATHEMATICS (Pure & Applied)
NB : All questions may be attempted.
Question 1. [12 marks]
Consider the following initial-value problem: (y 2 + 1)y 0 = 2ty, y(2) = 1.
(a) Discuss the existence/uniqueness of a solution to the above IVP. If possible, specify the
largest interval on which a solution exists and the largest interval on which it is unique.
(b) Find a solution to the IVP.
[7,5]
Question 2. [20 marks]
Find a general solution to the following differential equations. If an initial value is specified,
solve the IVP. You may leave the answer in implicit form, but simplify as much as possible.
(a) ty0 + 5y = 7t 2 , y(2) = 5;
(b) t 2y 0 + 2ty = 5y 3 ;
(c) y 000 − 3y 0 − 2 = 0.
[7,8,5]
Question 3. [8 marks]
Using the method of Variation of Parameters, find a particular solution yp to the equation
t 2 y 00 − 4ty0 + 6y = t 3 if yc = At2 + Bt3 is the general solution to the associated homogenous
equation. [8]
Question 4. [10 marks]
Compute the following Laplace transforms. You may use the attached table of transforms.
(a) L 3 sin(2t) + t 3 ;
(b) L t 2 e 3t ; (c) L n cosh(2t) t o .
[2,3,5]
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