SPECIAL INSTRUCTIONS
• Show and explain all of your work unless the question directs otherwise. Simplify all answers.
+ The use of a calculator is not permitted.
• Answer the questions in the spaces pr‹ivided on the q
...
SPECIAL INSTRUCTIONS
• Show and explain all of your work unless the question directs otherwise. Simplify all answers.
+ The use of a calculator is not permitted.
• Answer the questions in the spaces pr‹ivided on the question sheets. If you run out of room for an answer, esk for extra paper.
This is a closed-hook, individual exam.
Find the general solution ‹if 2. Show that when Euler’s rnetliotl is used to approximate the solution of the initial value problem
y’ = 5y, y(tl) = 1,
at z = 1, then the approximation with stepsize ft is (1 + rh) '"'
3. A brine solution flows at a constant rate of 8 L/min into a lnrge tank that initially held lfl() L of water in which was dissolveil 0.6 kg of salt. The solution inside the t‹ink is kept well stirred and flows out of the tank at a rate of 8 L/iriin. The solution entering the tank contains disso1ve‹1 salt at a concentration of 0.05 kg/L Let A(I) represent the mass of salt in the tank at time t minutes. Finrl A(t .tt,li 225 (At›ri1 1Ctli, 2t)1g) Final Exaiii @2, Inrlividiiiil Test
(b) When will the c‹iiicentration of salt in the tank reach 0.02 kg/L‘/
6. For each of the differential equations below, determine the form of the particular solution (do not evaluate the coefficients!). The homogeneous solution is given.
(a) y" -I- 9// 4/3 sin(3t) (yg(I) cv cos(3/) -I- sin(3t))
7. Find the particular solution tti the rlifferential equation 8. An 6 kg mass is attaclietl to a spring with spring constant 40 fi/m, rind the system is at rest. At t,inie t — I), an external force F(t) = ‹Nos(2f) N is applied to the system. The damping constant for t.he system is 3 Ns/in.
(o) Translate the wortl problem into an initial value problem.
4 S. Find the Laplace transfortli Of the functi‹in /i(t) = e°’I siri(2t). 10. Theory
(a) Use Tlieoretii 6 (see the A‹lclitiona1 Information pages si'1'rlied with this exam) to discuss the existence anrl uniqueness ‹if a solution to the initial value pt obleni
(1 -I- t‘*) ” -I- t¿’ — y - tait(/), v(/ ) = Vo. y’(t„) — Vt, where tt . anti are real constants
5 (b) Prove proper ty P4 (see the Additionnl Information provitled with this exam). 11. Use the method of Laplace Transforms to solve the iiiiti‹i1 vnlue problem
y” -I- 4y - g(/), t/(0) — — I, y’(U) 0,
where
AdcJitional lnforiiiation
Theorem S Suppose p(t), q(l), and g(I) are ‹ ontinuoiis on an interavl (‹i, 6) that contains the point fb. Then, for any choice of the initial values TQ and Yt, there exists a initi ie solution y(t) on the same interval (n, b) to the initial value problem
^”(i) + n(i)»’(t) + 4( )v(^) = »(^)• y(to) = *'•• v’(fo) = *'i-
BRIEF TABLE OF LAPLACE TRANSFORMS
BRIEF TABLE OF PItOPERTIES OF THE LAPLACE TRANSFORM I
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