Sample Midterm ENGR 213 lecture P onlySeptember 2007

Instructor: Dr. Pawe l Gora

Time allowed: 1h 15min.

Material allowed: calculators.

Recommendations: use only blue or black ink

[10 points] Problem 1.Find the general solution of the following ODE

dy

dx=

x2

y(1 + x3).

[10 points] Problem 2.Determine which of the following ODE’s is exact and then solve it (in implicit form).

(a) (y cos x + 2xey)dx + (sin x + x2ey − 1)dy = 0

(b)(

3xy + y2)

dx +(

x2 + xy)

dy = 0

[10 points] Problem 3.(i) Which of the following two first order equations is linear? Explain why the other is not linear

(a) (1 + t2)dy

dt+ 4ty = (1 + t2)−2

(b)dy

dx− x =

1

y

(ii) Find the general solution of the linear equation that you have found above.

[10 points] Problem 4.Perform the proper substitution in the following Bernoulli ODE so as to obtain a new linear ODE and solve:

t2dy

dt+ 2ty − y3 = 0 .

[10 points] Problem 5.A ball with mass 0.15 kg is thrown upward with initial velocity 20 m/sec from the roof of a building 30 m high.Assume air resistance of |v|/30, where the velocity is measured in m/sec.

(a) Find the maximum height above the ground that the ball reaches.(b) Find the time that the ball hits the ground.

[10 points] Problem 6.Solve the following equation by using a substitution of the form u = Ax + By + C

(x + 2y)y′ = 1 , y(0) = −1 .

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