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Prof. S. Brick Differential Equations Exam 1 Math 238 Spring ’06 section 101

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Show all of your work, and explain your reasoning.

1. Solve y′ = xy2 + x

2. Draw the phase diagram of the autonomous differential equation y′ = y(5− y)(y + 2)2 and use it to sketch solution curves. (Here x ≥ 0, but y may be negative.) Identify and classify the types of equilibrium solutions.

3. The vertical motion of an object near the surface of the Earth is subject to two forces: a downward gravitational force FG = −mg (where m is the mass of the object, g is the gravitational constant, and the negative sign represents the downward direction) and a force FR due to air resistance. Assuming that the force due to air resistance is proportional to the velocity and using the fact that the sum of the forces is equal to the product of mass and acceleration, set up a differential equation for the velocity. Mention the signs of any other constants you use, explaining your reasoning.

4. The function y = Ce−x + x− 1, where C is a constant, is a solution to y′ = x− y (you need not check that). Find C such that the initial condition y(0) = 10 is satisfied.

5. Solve y′ + 2y x

= √

x.

6. Solve ( cos(x) + ln(y)

) dx +

(√ y +

x

y

) dy = 0. You may give an implicit form for your

solution.

7. Using an appropriate substitution, transform xy′+7x3y = (x2 +9)y 1 3 into either a first

order linear equation or a separable equation. Explicitly mention the substitution and whether the result is linear or separable. Do not solve.

8. Using an appropriate substitution, transform homogeneous equation xy2y′ = x3 + y3. into either a first order linear equation or a separable equation. Explicitly mention the substitution and whether the result is linear or separable. Do not solve.