Math 2280
Quiz 1
−x
1. Verify that for any C the function y = Ce + x − 1 is a solution to the equation y ′ = x − y, and find the
solution satisfying the initial condition y(0) = 10.
In order to check that y is a solution, we need to compute y ′ . We have
y ′ = −Ce−x + 1 = −Ce−x + x − x + 1 = x − (Ce−x + x − 1) = x − y,
so y is indeed a solution. In order to satisfy y(0) = 10, we must have
10 = y(0) = C − 1 ⇒ C = 11,
so our particular solution is
y(x) = 11e−x + x − 1.
2. At noon a car starts from rest at a point A and proceeds at constant acceleration along a straight road toward
point B. If the car reaches B at 12:50 pm with a velocity of 60 mi/hr, what is the distance from A to B?
Here we can use the equation
1 2
at + v0 t + x0
2
derived in class. We have v0 = 0, and we can take the point A to be at x = 0, which gives
x(t) =
x(t) =
1 2
at ⇒ v(t) = at.
2
The time to complete the trip is 50 minutes, or 5/6 hour, so
60 = a
5
⇒ a = 72.
6
Knowing the acceleration, we can compute
x
2
5
1
25
5
= (72)
= 36 ·
= 25 mi.
6
2
6
36
1