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Williams-Spring2010-Math108-M1-Solutions

# Williams-Spring2010-Math108-M1-Solutions - Math 108 —...

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Unformatted text preview: Math 108 — Williams —Midterm 1 - 2/17/2010 60 Minutes; no notes, books, calculators; you may use the ofﬁcial formula sheets. Name: Solm‘ms Section 03 / Section 04 To receive credit, you must show your work, explain your reasoning, and demonstrate the use of the speciﬁc method indicated, when one is indicated. Clarity willgbe considered in grading. ' “I have adhered to the Duke‘Community Standard in completing this examination.” Signature: .APJN Total (out of 24): 1. (6 points) Find the genera} solution of y" +43; = 3x3_ ‘3“+ﬁr 13H 60W~<v> L3” +4520, (Lam Egg ram :0; n31: 4:23, an '— 6.. (30‘4”) + again (ﬁx) , m AX3+ sz'l' Cx +D- MM"? \$6313sz %:+Lh3; 3A2— .3?) : 3pm1 +- 29x +C— _ 1g?" 5 (Hind-2E) (okXi'lB +4CAX3+bxz+cx+m= Bﬁ EM (192.993.? W . “4%: Lie, 44/ g, C: “cl/g mm- W ##M , (3103605sz3 “' C2 \$rﬂL2K) + 94 x3 Cy? X] 2. A. (3 points) Find an integrating factor it = “(2.1) and show that it transforms this equation into an exact equation (dg not solve): (x2 + y2)dx — xy dy = O W aﬁﬁwwzv = law—mm at): -3 — 33M — 3% J14 -31”le = ”bl/Ll “‘1!le = be] i (I mtg HAM—0‘ ea ial'ﬁﬁmlvj gala) W5 ’ (x) 0% ”Mg-z + 52g) .3. 3/34 342) 2 ._ B. (2 points) Find a differential equation whose general solution is given by sin(x+y)+y3+y2=c 3. (5 points) Knowing that 321 = x2 is a soiution to x231": — 2y = 0, use the method of reduction of order to ﬁnd a second solution to this equation. £217 51; VLXNé, -_-. votwcz, 7m 57: -:.. v’w'ir 21v 5'2." :— v”x" + ZCZX v") +7~V Bing up 40 ODE : v’y‘qqﬁvH-ivwz- Six/x” =0 50‘1'" _ V”(3¢‘f) + Wax-i) + v61xl—Zx") “‘0 DIV/"6Q 0741’ 363: Xv”+Lf\/’=0. (tail 2:5”- x7z’+ 4;. =0 xdy¢m¢*‘fa MMWLQ') J; 012 = "if; Ax mm —. mixi“i +C Z- = CXHf = v’ => \/=‘&>€_3. i , -3 i- , z: _. 4. (4 points) Show that the substitution v = ax + by + c transforms the differential equation g:- = ﬂax + by + c) into a separable equation. “*5/01 ‘90!) )6 7' ’ NM Adj/aw 139'» W5 5%: v. V: oLaé-i-bopc, a»; ()8; v—c-atxp 5. (4 points) Use the Laplace Transform to solve y' = 60: — a). ...
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