Differential Equations, Math 441, Homework 1
Due: Friday January 27, in the beginning of the class.
Problem 1: [Variation of constants] (12 points)
Consider the differential equation
y0=ay +b
for some contants a, b ∈Rand initial condition y(0) = y0.
We know a solution for the differential equation
y0
1=ay1,
namely y1(t) = eat. Make the ansatz y(t) = v(t)y1(t).
i) Find a differential equation for v
ii) Solve the differential equation for v.
iii) Use this to find the solution for the differential equation y0=ay +b.
Problem 2: (10 points) Let I= (α, β) be an open interval and p:I→R
a continuous function. Show that for any choice of t0∈I, any solution of
the equation
µ0=pµ on I
is a constant multiple of the function exp(Rt
t0p(s)ds).
The following problems are from Boyce and DiPrima, 8th edition. Hand in
only the problems marked with an asterix (*). Each problem is worth 5
points.
Sec. 1.1: Problems 7*, 17*, 19*, 20.
Sec. 1.2: Problem 19*.
Sec. 1.3: Problems 1*, 3*, 5, 19.
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