Math 34 Differential Equations Exam #1 October 9, 2003 SHOW ALL WORK Solve the following differential equations

[15] 1a.) t^{2}y′ + 2ty = t sin(t).

Answer 1a.)

[15] 1b.) y′′ − 4y′ + 4y = 0, y(0) = 2, y′(0) = 3.

Answer 1b.)

[15] 1c.) y′′ − 4y′ + 4y = 5t + 1.

Answer 1c.)

[15] 1d.) 2y′′ − 3y^{2} = 0, y(0) = 1, y′(0) = 1..

Answer 1d.)

[5] 2.) Use Euler’s formula to write e^{2+3i} is the form of a + ib.

Answer 2.)

[5] 3.) Draw the direction field for y′ = y.

4.) Circle T for true and F for false.

[3] 4a.) Suppose ψ_{1} and ψ_{2} are solutions to the linear equation, ay′′ + by′ + cy = g(t), then ψ_{1} + ψ_{2} must also be a solution to ay′′ + by′ + cy = g(t).

T F

[3] 4b.) Suppose ψ_{1} and ψ_{2} are solutions to the equation, ay′′ + by′ + cy^{2} = 0, then ψ_{1} + ψ_{2} must also be a solution to ay′′ + by′ + cy^{2} = 0.

T F

[3] 4c.) Suppose ψ_{1} is a solution to the linear equation, ay′′ + by′ + cy = g(t), and ψ_{2} is a solution to the linear equation, ay′′ + by′ + cy = f(t), then 5ψ_{1} + 3ψ_{2} must also be a solution to ay′′ + by′ + cy = 5g(t) + 3f(t).

T F

[3] 4d.) If p, q, and g are continuous, then there exists a unique solution to

y′′ + p(t)y′ + q(t)y = g(t), y(t_{0}) = y_{0}, y(t_{1}) = y_{1}. T F

[3] 4e.) If p, q, and g are continuous, then there exists a unique solution to

y′′ + p(t)y′ + q(t)y = g(t), y(t_{0}) = y_{0}, y′(t_{0}) = y′_{0}. T F

[3] 4f.) Given an initial value, there always exists a unique solution to any first order differential equation.

T F

5.) Choose one of the two following problems. Clearly indicate which problem you have chosen.

5A.) Suppose the equation \frac{dp}{dt} = γp describes the population of field mice. If the population of field mice doubles in 10 years, how long will it take the population to quadruple.

5B.) Find the escape velocity for a body projected upward with an initial velocity v_{0} from a point 3R above the surface of the earth, where R is the radius of the earth. Neglect air resistance. Recall that the equation of motion is m\frac{dv}{dt} = −\frac{mgR^{2}}{(R+x)^{2}} where x is the distance from the earth’s surface.

Answer 5A or 5B.)