Math 2280-1
Quiz 1
January 16, 2015
1a) Consider the differential equation for y = y x :
y#= y K x
x
Show that the functions y x = x C 1 C C e solve this differential equation (Where C = constant).
(4 points)
We verify that the given functions make the differential equation a true equation. So we compute the left
and right sides of the equation and verify that they are equal:
d
LHS: y# x =
x C 1 C C ex = 1 C C ex .
dx
RHS: y x K x = x C 1 C C ex K x = 1 C C ex.
Thus LHS=RHS for these functions y x , and they solve the DE.
1b) Use the solutions in (a) to find a solution to the initial value problem
y#= y K x
y 0 =0
(3 points)
x
x
For y x = x C 1 C C e we want y 0 = 0, i.e. 1 C C = 0 0 C =K1 0 y x = x C 1 K e .
1c) Sketch the graph of the solution function to part (b) onto the slope field below.
(3 points)
It will be the graph going through the initial point 0, 0 , and each point of the graph will have the same
slope as indicated by the direction field.