8. [12 Points] Find a particular solution of the differential equation. t 2 y + ty 4y = t 3, y h = c 1 t 2 + c 2 t 2.

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1 Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will not be given for anwer (even correct one) without upporting work. A table of Laplace tranform, a table of convolution product, and the tatement of the main partial fraction decompoition theorem have been appended to the exam. In Exercie 7, olve the given differential equation. If initial value are given, olve the initial value problem. Otherwie, give the general olution. Some problem may be olvable by more than one technique. You are free to chooe whatever technique that you deem to be mot appropriate.. [ Point] y + y = 3e t e 3t, y(0) =.. [ Point] y = y y, y(0) =. 3. [ Point] y + 5y + y = 0, y(0) =, y (0) =. 4. [ Point] 4t y + y = [ Point] 4y + 9y = 0, y(0) =, y (0) = [ Point] y + 6y + 9y = e 3t. 7. [ Point] y + y = 3h(t π), y(0) =, y (0) = 3. Recall that h(t) refer to the unit tep function. 8. [ Point] Find a particular olution of the differential equation t y + ty 4y = t 3, given the fact that the general olution of the aociated homogeneou equation i y h = c t + c t. 9. [ Point] Find the Laplace tranform of each of the following function. t if 0 t <, (a) f(t) = 8 t if t < 4 0 if t 4. (b) g(t) = te t co 3t 0. [ Point] Compute each of the following invere Laplace tranform. { } + 5 (a) L { } (b) L 4 ( + )( 4) Math 065 Section December 0, 008

2 . [8 Point] Let A = [ ] 7 8. (a) Compute (I A). (b) Find L {(I A) }. (c) Find the general olution of the ytem y = Ay. [ ] (d) Solve the initial value problem y = Ay, y(0) =.. [ Point] A tank initially contain 500 gallon of water in which 0 pound of alt i initially diolved in the water. Brine (a water-alt mixture) containing 0.5 pound of alt per gallon flow into the tank at the rate of 4 gal/min, and the mixture (which i aumed to be perfectly mixed) flow out of the tank at the ame rate of 4 gal/min. (a) Find the amount of alt y(t) in the tank at time t. (b) How much alt doe the tank contain after hour? (c) What i lim t y(t)? Math 065 Section December 0, 008

3 A Short Table of Laplace Tranform. L {af(t) + bg(t)} () = af () + bg(). L {e at f(t)} () = F ( a) 3. L {f(t c)h(t c)} = e c F () 3. L {g(t)h(t c)} = e c L {g(t + c)} 4. L { tf(t)} () = d d F () 5. L {f (t)} () = F () f(0) 6. L {f (t)} () = F () f(0) f (0) { } t 7. L f(x) dx () = F () 0 8. L {(f g)(t)} () = F ()G() 9. L {} () = 0. L {t n } () = n! n+. L {e at } () =. L {t n e αt } () = 3. L {co bt} () = 4. L {in bt} () = 5. L {e at co bt} () = 6. L {e at in bt} () = 7. L {h(t c)} () = e c a n! ( α) n+ + b b + b a ( a) + b b ( a) + b Math 065 Section December 0, 008 3

4 Table of Convolution f(t) g(t) f g(t). t t n t n+. t in at 3. t in at 4. t co at (n + )(n + ) at in at a a (co at ( a t )) 3 co at 5. t co at (at in at) a3 6. t e at e at ( + at) a 7. t e at a 3 (eat (a + at + a t )) 8. e at e bt b a (ebt e at ) 9. e at e at te at 0. e at in bt. e at co bt. in at in bt 3. in at in at 4. in at co bt 5. in at co at 6. co at co bt 7. co at co at a a b a + b (beat b co bt a in bt) a + b (aeat a co bt + b in bt) (b in at a in bt) b a a b (in at at co at) a (a co at a co bt) b a a b t in at (a in at b in bt) a b a b (at co at + in at) a Math 065 Section December 0, 008 4

5 Partial Fraction Expanion Theorem The following two theorem are the main partial fraction expanion theorem, a preented in the text. Theorem (Linear Cae). Suppoe a proper rational function can be written in the form p 0 () ( λ) n q() and q(λ) 0. Then there i a unique number A and a unique polynomial p () uch that p 0 () ( λ) n q() = A ( λ) n + p () ( λ) n q(). () The number A and the polynomial p () are given by A = p 0() q() and p () = p 0() A q(). () =λ λ Theorem (Irreducible Quadratic Cae). Suppoe a real proper rational function can be written in the form p 0 () ( + c + d) n q(), where + c + d i an irreducible quadratic that i factored completely out of q(). Then there i a unique linear term B + C and a unique polynomial p () uch that p 0 () ( + c + d) n q() = B + C ( + c + d) n + p () ( + c + d) n q(). (3) If a + ib i a complex root of + c + d then B + C and the polynomial p () are given by B + C =a+bi = p 0() q() and p () = p 0() (B + C )q(). (4) =a+bi + c + d Math 065 Section December 0, 008 5