Math 34 Differential Equations Exam #2 October 29, 2010 SHOW ALL WORK

1a.) L(0) =

1b.) L^{−1}(\frac{2}{(s−4)^{2}+5}) =

[4] 2.) Circle T for True or F for False:

[4] Suppose y = f(t) is a solution to 3y′′ + 10y = cos(t), y(0) = 0, y′(0) = 0 and suppose y = g(t) is a solution to 3y′′ + 10y = cos(t), y(0) = 100, y′(0) = −200.

For large values of t, f(t) − g(t) is very small.  T  F

3a.) Given 2y′′ + 5y = cos(wt), determine the value w for which undamped resonance occurs: \underline{w =}

3b.) Briefly describe in words the long-term behaviour of a solution to 2y′′ + 5y = cos(wt) for this value of w.

[15] 4.) A mass of 4 kg stretches a spring 5m. The mass is acted on by an external force of 6e^{t} N (newtons) and moves in a medium that imparts a viscous force of 8 N when the speed of the mass is 15 m/sec. The mass is pulled downward 1m below its equilibrium position, and then set in motion in the upward direction with a velocity of 10 m/sec.

Formulate the initial value problem describing the motion of the mass.

[25] 4.) Use ch 3 methods to solve the given initial value problem.

y′′ + 4y = sin(t), y(0) = 0, y′(0) = 0

1.) Find the general solution to y′′ + 4y = 0: r^{2} + 4 = 0 implies r = ±2i

homogeneous solution: y(t) = c_{1}cos(2t) + c_{2}sin(2t)

2.) Find ONE solution to y′′ + 4y = sin(t): Educated guess: y = Asin(t) (since no y’ term). y′ = Acos(t)

ψ_{i}

This includes plugging guessed solution into ay′′ + by′ + cy = g_{i} to find constant(s). The general solution to [**] is

c_{1}ϕ_{1} + c_{2}ϕ_{2} + ψ_{1} + ψ_{2} + ...ψ_{n}

3.) If initial value problem: Once general solution is known, can solve initial value problem (i.e., use initial conditions to find c_{1}, c_{2}).

[28] 5.) Use the LaPlace transform to solve the given initial value problem. y′′ + 4y = sin(t), y(0) = 0, y′(0) = 0 [15] 6.) Section 6.2 Thm 6.2.1: Hint: use integration by parts and let dv = f′(t) Cor 6.2.2’: L(f′′(t)) = s^{2}L(f(t)) − sf(0) − f′(0) Hint. Use that f′′ is the derivative of f′. Let g = f′ and note L(g′(t)) = sL(g(t))− g(0) Table 6.2.1 Section 6.3 Thm 6.3.1 Hint: ∫_{0}^{∞} h(t)dt = ∫_{0}^{c} h(t)dt + ∫_{c}^{∞} h(t)dt and use u-substitution (let u = t − c). Thm 6.3.2 Hint: Let F (s) = L(f(t)). Use definition of Laplace transform to evaluate L(e^{ct}f(t)) and F (s − c).