Math 334
Assignment 3
Due: 12 Noon on Thursday, October 5, 2006.
1. Let φ1(x) and φ2(x) be linearly indep endent solutions of the equation y′′ +P(x)y′+Q(x)y= 0 on
(a, b).
(a) Show that they can not b oth be zero at the same p oint x0∈(a, b).
(b) Show that they can not b oth have an extremum at the same point x0∈(a, b).
2. Consider the following differential equation:
(x−1)y′′ −y′= 0.
(a) Find the general solution of this equation.
(b) Evaluate W[φ1, φ2](1) where φ1and φ2are linearly independent solutions of the equation and W
is the Wronskian. How do you reconcile this with the theorem in the notes?
3. Find the general solution to the following homogeneous equations:
(a) y′′ + 5y′+ 6y= 0;
(b) y′′ +y′−y= 0;
(c) y′′ + 9y= 0;
(d) y′′ −6y′+ 10y= 0.
4. Find the solution to the following initial value problems:
(a) y′′ + 2y′+y= 0, y(0) = 1, y ′(0) = −3;
(b) y′′ −2y′−2y= 0, y(0) = 0, y ′(0) = 3;
(c) y′′ −2y′+ 2y= 0, y(π) = eπ, y ′(π) = 0.
5. Consider the following initial value problem:
y′′ +βy′+ 4y= 0, y (0) = 1, y′(0) = 0.
(a) Solve the problem for β= 5.
(b) Solve the problem for β= 4.
(c) Solve the problem for β= 2.
(d) Sketch the solutions obtained in parts (a), (b), and (c).
6. Consider the differential equation y′′ −λ2y= 0, where λis a positive constant.
(a) Show that the general solution can be written in the form c1eλx +c2e−λx .
(b) Show that the general solution can also be written in the form a1cosh λx +a2sinh λx.
(c) Use each of these general solution formats to solve the initial value problem:
y′′ −λ2y= 0, y(0) = α0, y′(0) = α1.
Which is more convenient?
