Problem 11
Solve the differential equation.
$ 2 \dfrac{d^2y}{dt^2} + 2 \dfrac{dy}{dt} - y = 0 $
Problem 12
Solve the differential equation.
$ \dfrac{d^2R}{dt^2} + 6 \dfrac{dR}{dt} + 34R = 0 $
Problem 13
Solve the differential equation.
$ 3 \dfrac{d^2V}{dt^2} + 4 \dfrac{dV}{dt} + 3V = 0 $
Problem 14
Graph the two basic solutions along with several other solutions of the differential equation. What features do the solutions have in common?
$ 4 \dfrac{d^2y}{dx^2} - 4 \dfrac{dy}{dx} + y = 0 $
Problem 15
Graph the two basic solutions along with several other solutions of the differential equation. What features do the solutions have in common?
$ \dfrac{d^2y}{dx^2} + 2 \dfrac{dy}{dx} + 2y = 0 $
Problem 16
Graph the two basic solutions along with several other solutions of the differential equation. What features do the solutions have in common?
$ 2 \dfrac{d^2y}{dx^2} + \dfrac{dy}{dx} - y = 0 $
Problem 17
Solve the initial-value problem.
$ y" + 3y = 0 $, $ y(0) = 1 $, $ y'(0) = 3 $
Problem 18
Solve the initial-value problem.
$ y" - 2y' - 3y = 0 $, $ y(0) = 2 $, $ y'(0) = 2 $
Problem 19
Solve the initial-value problem.
$ 9y" + 12y' + 4y = 0 $, $ y(0) = 1 $, $ y'(0) = 0 $
Problem 20
Solve the initial-value problem.
$ 3y" - 2y' - y = 0 $, $ y(0) = 0 $, $ y'(0) = -4 $
Problem 21
Solve the initial-value problem.
$ y" - 6y' + 10y = 0 $, $ y(0) = 2 $, $ y'(0) = 3 $
Problem 22
Solve the initial-value problem.
$ 4y" - 20y' + 25y = 0 $, $ y(0) = 2 $, $ y'(0) = -3 $
Problem 23
Solve the initial-value problem.
$ y" - y' - 12y = 0 $, $ y(1) = 0 $, $ y'(1) = 1 $
Problem 24
Solve the initial-value problem.
$ 4y" + 4y' + 3y = 0 $, $ y(0) = 0 $, $ y'(0) = 1 $
Problem 25
Solve the boundary-value problem, if possible.
$ y" + 16y = 0 $, $ y(0) = -3 $, $ y(\pi/8) = 2 $
Problem 26
Solve the boundary-value problem, if possible.
$ y'' + 6y' = 0 $, $ y(0) = 1 $, $ y(1) = 0 $
Problem 27
Solve the boundary-value problem, if possible.
$ y'' + 4y' + 4y = 0 $, $ y(0) = 2 $, $ y(1) = 0 $
Problem 28
Solve the boundary-value problem, if possible.
$ y'' - 8y' + 17y = 0 $, $ y(0) = 3 $, $ y(\pi) = 2 $
Problem 29
Solve the boundary-value problem, if possible.
$ y'' = y' $, $ y(0) = 1 $, $ y(1) = 2 $
Problem 30
Solve the boundary-value problem, if possible.
$ 4y'' - 4y' + y = 0 $, $ y(0) = 4 $, $ y(2) = 0 $
Problem 31
Solve the boundary-value problem, if possible.
$ y" + 4y' + 20y = 0 $, $ y(0) = 1 $, $ y(\pi) = 2 $
Problem 32
Solve the boundary-value problem, if possible.
$ y" + 4y' + 20y = 0 $, $ y(0) = 1 $, $ y(\pi) = e^{-2\pi} $
Problem 33
Let $ L $ be a nonzero real number.
(a) Show that the boundary-value problem $ y'' + \lambda y = 0 $, $ y(0) = 0 $, $ y(L) = 0 $ has only the trivial solution $ y = 0 $ for the cases $ \lambda = 0 $ and $ \lambda < 0 $.
(b) For the case $ \lambda > 0 $, find the values of $ \lambda $ for which this problem has a nontrivial solution and give the corresponding solution.
Problem 34
If $ a $, $ b $, and $ c $ are all positive constants and $ y(x) $ is a solution of the differential equation $ ay'' + by' + cy = 0 $, show that $ \lim_{x \to \infty} y(x) = 0 $.
Problem 35
Consider the boundary-value problem $ y''- 2y' + 2y = 0 $, $ y(a) = c $, $ y(b) = d $.
(a) If this problem has a unique solution, how are $ a $ and $ b $ related?
(b) If this problem has no solution, how are $ a $, $ b $, $ c $, and $ d $ related?
(c) If this problem has infinitely many solutions, how are $ a $, $ b $, $ c $, and $ d $ related?
