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Material Type: Exam; Class: Differential Equations; Subject: Mathematics; University: Colorado School of Mines; Term: Fall 2008;
Typology: Exams
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MATH 225 - June 5, 2008 NAME: Exam III - 50 minutes - 50 Points
In order to receive full credit, SHOW ALL YOUR WORK. Full credit will be given only if all reasoning and work is provided. When applicable, please enclose your final answers in boxes.
(b) y′′^ − y = 0
(c) y′′^ = 0
(b) Write down the form of the particular solution supposing that f (t) is given by: i. f (t) = 2e−t ii. f (t) = 3 iii. f (t) = 5e−^2 t iv. f (t) = 3 cos(3t)
do not solve for the unknown constants. if using imaginary exponentials be sure to include whether the real or imaginary part should be kept.
2 y′′^ − 8 y = 16 − 18 e−t, y(0) = 1, y′(0) = − 3 (1)
y′^ + 2y = 0. (2)
(a) Assume a power-series solution to the ODE and find the corresponding recurrence relation for the power-series coefficients.
(b) Solve the recurrence relation for these coefficients and using a known Taylor series find a transcendental expression for your solution.
(c) Check your result.
y′′^ + y = 0, y(0) = − 1 , y′(0) = 1. (4)
(a) Convert the second-order ODE into a system of first order ODE’s.
(b) Using eigenvalues and eigenvectors, solve the corresponding initial value problem. Express you solution in real form.