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Math 340 Elementary Differential Equations Exam 1
Closed book. You may use a calculator and one 8 1 /2 × 11 sheet of handwritten notes (both sides). You must show your work to receive full credit. In problems where you are asked to justify your answer, you will be graded on both correctness and clarity. Errors in spelling and grammar may reduce the clarity of your answer.
1. (a) (6 pts) For each of the equations below, indicate its order and determine whether it is autonomous (Yes or No) and whether it is linear (Yes or No).
equation order autonomous? linear?
cos(xy)y ’’’ − 3e x y = sin(y) −8y ’’ + 3y’ − 2y = 0 cos(xy)y ’’’ − 3e x y = sin(x)
(b) (4 pts) Find values for a and b so that
y = 2xeax + b is a solution to
y ’’ + 4y’+ 4y = 7.
2. (10 pts) Find all solutions to the differential equation y’= (y + 1)e x .
3. (10 pts) If z(t) solves the differential equation z 0 + cos(t)z = te− sin(t) , z(0) = 2, what is z(π)?
4. Consider the differential equation y’− y x = − sin x x y 2 .
(a) (6 pts) Change variables with v = y −1 and find a linear differential equation for v.
(b) (2 pts) Show that v = c−cos(x) x solves the differential equation in (a) for any c.
(c) (2 pts) Find all solutions to the original differential equation in y. (Don’t forget to check for singular solutions!)
5. (10 pts) Match each of the following 4 differential equations with its corresponding slope field on the following page. Justify your answer.
1. y’ = e x
2. y’= e x−y
3. y’= e y
4. y’= e x − e y
6. Consider the differential equation y’ = y(1 − y)(1 − e −y ).
(a) (4pts) Draw the phase diagram.
(b) (4 pts) Sketch the slope field along with typical solutions.
(c) (2 pts) If y solves the differential equation and y(1) = 2, what is limx→∞ y(x)?
7. (10 pts) For the initial value problem y’ = − cos(x − y), y(0) = 2, use Euler’s method with a step size of 0.1 to approximate y(0.3). (Compute all steps with at least 4 digits of decimal precision.)
8. Consider the differential equation d 2 y dx2 + dy dx2 = 0 using the change of variables v = e y .
(a) (3 pts) Use the chain rule to write dv dx in terms of y and dy dx .
(b) (4 pts) Show that d 2 v dx2 = e y d 2 y dx2 + dy dx2 ! .
(c) (3 pts) Use part (b) to solve the differential equation for y.