MATH 235 - Differential Equations w/ Honors January 14, 2008
Homework 2, Spring 2008 Due: January 21, 2008
1. Solutions to ordinary differential equations.
(a) Show that y(t) = e2t+cet, where c∈R, is a solution to dy
dt −y=e2t.
(b) Show that x2+y2=cx, where c∈R, is a solution to 2xy dy
dx =y2−x2.
(c) Show that y(t) = c1sinh(t) + c2cosh(t), where c1, c2∈R, is a solution to y00 −y= 0.
(d) Show that y(t) = c1sin(t) + c2cos(t), where c1, c2∈R, is a solution to y00 +y= 0.
(e) Show that x(t) = Ae−k1tand y(t) = k1A
k1−k2
e−k2t+k1A
k2−k1
e−k1tare solutions to the system of differential
equations.
dx
dt =−k1x, x(0) = A(1)
dy
dt =k1x−k2y, y(0) = 0 (2)
Hint: For problem 1c recall that sinh(t) = et−e−t
2,cosh(t) = et+e−t
2.
2. Solve the following problems via separation of variables. When appropriate solve for the integrating constant C
using the initial value which is given.
(a) dy
dt = 1 + 1
y2.
(b) (y0)2−xy0+y= 0.
(c) dy
dt = (y2+ 1)t, y(0) = 1.
(d) dy
dt =yet
1 + y2.
Hint: For (b) consider completing the square and using the variable substitution z=−(y−t2/4).
3. Section 1.3 of the text, problems 8, 10, 15.
4. Consider the polynomial p(y) = −y3−2y+ 2.
(a) Using HPGSolver sketch the slope field for dy
dt =p(y).
(b) Using HPGSolver, sketch the graphs of some of the solutions using the slope field.
(c) Describe the relationship between the roots of p(y) and the solutions of the differential equation.
(d) Using Euler’s method, approximate the real root(s) of p(y) to three decimal places.
5. The following nonlinear system has been proposed as a model for a predator-prey system of two particular
species of microorganisms.
dx
dt =ax −by√x(3)
dy
dt =cy√x, (4)
where a, b, c ∈R+. In this case the variables xand yare dependent variables and appear in both ODE’s and
thus the ODE’s are said to be coupled.
(a) Which variable, xor y, represents the predator population? Which variable represents the prey population?
Justify your choices.
(b) What happens to the predator population if the prey is extinct? Justify your conclusion.
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