MATH 1320 : Spring 2014
Lab 3
Lab Instructor : Kurt VanNess
Name:
Score:
Write all your solutions on a separate sheet of paper.
1. A function y(t) satisfies the differential equation
dy
= y 4 − 5y 3 + 6y 2
dt
(a) What are the constant solutions or equilibrium solutions for the equation?
(b) For what values of y is y increasing?
(c) For what values of y is y decreasing?
2. Use Euler’s method with step size 0.1 to estimate y(0.5), where y(x) is the solution of the initial-value
problem y 0 = x − y 2 , y(0) = 1.
3. (a) Solve the differential equation y 0 = x + y by making the change of variable u = x + y.
(b) Solve the differential equation xy 0 = y + xey/x by making the change of variable u = y/x.
4. Experiments show that the reaction H2 + Br2 −→ 2HBr satisfies the rate law
d[HBr]
= k[H2 ][Br2 ]1/2
dt
and so for this reaction the differential equation becomes
dx
= k(a − x)(b − x)1/2
dt
where x = [HBr] and a and b are the initial concentrations of hydrogen and bromine.
(a) Find x as a function of t in the case where a = b. Use the fact that x(0) = 0.
(b) If a > b, find t as a function of x. [Hint: In performing the integration, make the substitution
√
u = b − x.]
5. A sphere with radius 1 m has temperature 15◦ C. It lies inside a concentric sphere with radius 2 m
and temperature 25◦ C. The temperature T (r) at a distance r from the common centers of the sphere
satisfies the differential equation
d2 T
2 dT
+
= 0.
2
dr
r dr
Solve it to find an expression for the temperature T (r) between the spheres. [Hint: If we let S = dT /dr,
then S satisfies a first-order differential equation.]
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