MATH 2255, Fall 2020
Homework 4
Due Friday, September 25, 11:30am.
Please upload your homework on Carmen. Late homework is not accepted. I
encourage you to work with others on homework problems, but you must write
up your own solutions. Solutions must be presented clearly, or will be marked
down.
(1) Consider the ODE given by
y
0 = −y − 1, y(0) = 0.
(a) Determine yn(t) for each n, where the yn are the functions obtained by successive
approximation. You should guess yn, then prove by mathematical induction that
your guess is correct.
(b) Write down y∞(t) using elementary functions.
(c) Solve the ODE to obtain the actual solution y(t).
(d) Plot y1, . . . , y4, along with the actual solution y(t).
(2) Let f(t, y) be a function such that fy = ∂f/∂y is continuous in some rectangle D.
Show that there is a positive constant K such that
|f(t, y1) − f(t, y2)| ≤ K|y1 − y2|,
where (t, y1) and (t, y2) are any two points in D having the same t-coordinate. This
exercise proves that if fy is continuous, then the Lipschitz condition is true. [Hint:
Hold t fixed, and view f(t, y) as a function of y only. Then use the mean value
theorem on f, and choose K to be the maximum value of |fy| on D.]
(3) Solve the given difference equations in terms of the initial value y0, and describe the
behaviour of the solution as n → ∞.
(a) yn+1 = −0.9yn;
(b) yn+1 =
qn+3
n+1 yn;
(c) yn+1 = (−1)n+1yn.
(4) A college student borrows $8000 to buy a car. The lender charges the annual interest
rate of 10%. What monthly payment is required to pay off the loan in 3 years?
(5) Solve the following differential equations.
(a) y
0 =
x−e−x
y+e
y ;
(b) (1 + t
2
)y
0 + 4ty = (1 + t
2
)
−2
;
(c) (3x
2 − 2xy + 2) + (6y
2 − x
2 + 3)y
0 = 0.
