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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