UNIVERSITY OF EXETER DEPARTMENT OF MATHEMATICS MAT 1125, Advanced Calculus CA Test 11th March 1998, 9.05-9.55am Solutions should be handed in for all questions

1. Which of the following functions are solutions of the differential equation

y′′ − 4y′ + 4y = e^{x} (i) e^{x}; (ii) e^{2x}; (iii): e^{2x} + e^{x}; (iv): xe^{x}.

[5 marks]

2. Find the general solution of the following differential equations:

(a)

(xy + 2y)\frac{dy}{dx} − (x^{2} + x^{2}y^{2}) = 0

[10 marks]

(b)

y′′ + y′ − 2y = e^{x}.

[10 marks]

3. (a) Evaluate the following limit:

lim_{x→0} \frac{ln(cos x)}{sin x}.

(b) Find the Maclaurin series for f(x) = sinh x.

[10 marks]

4. Test for convergence the following series clearly stating in each case which test

you are using.

(i)  ∑_{n=0}^{∞} \frac{n + 5}{3n^{4} + 7};
(ii)  ∑_{n=0}^{∞} \frac{2^{n} + 5}{3^{n}};
(iii)  ∑_{n=1}^{∞} \frac{1}{n^{ln 2}} .

[10 marks]

5. Find the radius of convergence of the series:

∑_{n=1}^{∞} \frac{(−1)^{n}x^{n}}{2^{n}n(n + 1)}

[5 marks]

Advanced Calculus Test 11.03.98