MATH 231, SPRING 2008- Homework Set 2.
For the following initial-value problems (1 to 4), find the solution, state
the interval of definition, sketch the graph (assume y=y(t) in all cases):
1. y0−3
ty=t4, y(1) = 1.
2. y0+2t
1+t4y= 0, y(0) = 1. (Why should one expect a positive
asymptotic value even before solving the equation?)
3. y0+ (2 + cos 3t)y= 0, y(0) = 1.
4. y0+ 3t2y=t2, y(0) = 1.
5. y0+ (cos 3t)y= cos 3t. (Find the general solution.) Are all solutions
periodic?
6. Solve y0+ (cos 3t)y= 2, y(0) = 1.Write the solution in the form:
y(t) = yper(t) + yb(t)t,
where yper(t) and yb(t) are periodic (that is, find yper(t) and yb(t).)
7. Show that the function F(t) defined by the integral:
F(t) = Zt
0
e−sin 5ssin 3sds
can be written in the form: F(t) = G(t) + A
2πt, where G(t) is 2π-periodic
and Ais a positive constant (find the numerical value of A.)
Hint: Use change of variables to show that:
F(t+ 2π) = F(t) + A, where A=Z2π
0
e−sin 5ssin 3sds,
then show that F(t)−A
2πtis 2π-periodic (as done in class on 1/24).
8. Solve the initial-value problem: y00 + 4y= 0, y(0) = 1, y 0(0) = 3.Give
the answer in the form Asin(2t+ϕ) (that is, find A > 0 and ϕ.)
9. Solve the initial-value problem: y00 −9y= 0, y(0) = 5, y 0(0) = 7.Give
the answer as a linear combination of sinh3tand cosh 3t.
10. Show that any function of the form:
y(t) = ae3t+be−3t
is zero for at most one value of t(unless the constants aand bare both
zero.) Hint: writing down the equations y(t1) = 0 and y(t2) = 0 gives a
linear system for aand b; when t16=t2, show the only solution is a=b= 0.
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