Homework Assignment 6 in MATH 308-Fall 2015
due September 30, 2015
Topics covered : linear homogeneous equations of second order with constant coefficient: the cases of real roots
and repeated roots (section 3.1 and section 3.4), review of complex numbers (beginning of section 3.3 and he first
item of the “Lecture notes and enrichment” in my homepage)
1. (a) Find the solution of the initial value problem
2y 00 − 7y 0 + 6y = 0,
y(0) = α, y 0 (0) =
1
4
(1)
where α is a parameter.
(b) Determine all values of α, if any, for which the solution of the initial value problem
(1) tends to +∞ as t → +∞.
2. Consider the differential equation 9y 00 + 24y 0 + 16y = 0.
(a) Find the general solution of this equation;
(b) Find the solution of this equation satisfying the initial conditions y(0) = 2, y 0 (0) = α;
(c) For the solutions obtained in the previous item find the values of α , if any, for which
the solutions tends to +∞ as t → −∞ and the values of α, if any, for which the
solutions tend to −∞ as t → −∞.
3. (a) Write the given expressions in the form a + ib:
i. (4 − 3i)(5 + 3i)
1+7i
ii. 2−8i
(Hint: multiply both numerator and denominator by complex conjugate of the denominator)
(b) Use Euler’s formula to write the given expression in the form a + ib:
5π
i. e4+ 3 i ;
ii. (−1 + i)11 .
1