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Unformatted text preview: MATHEMATICS 201, SECTION A01
MIDTERM 1
June 8, 2011 Name: Student Numb er: 0 Duration: 50 min. Maximum score: 30.
o The test has 6 problems and 6 pages.
0 Full answers, please. Justify your statements. 0 The Sharp EL—510R scientiﬁc calculator is the only calculator allowed. No other aids
such as books, notes or formula sheets are permitted. 1. 1. State the order of the diﬁerential equation and give reason(s) for Whether it is linear
or nonlinear. (a) y’” + 3:421” + 2y’ = 3x + 1 ﬂin/ Jig/fr" nan  ﬁnd (60 Mr, WM 3'3 “6 )
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V'VLw {J (b) y” + H22 y’ — (sin ﬂy = em 1 3L6 Mﬁ/ rJY‘Cl/u / C/. 11,. 5'3“.C§L ix, Midterm 1, May 2011 MATH 201 Page 2 2. For the differential equation 5: = y2(y + 1)(y — 2) (a) Find the critical points, if any7 and draw the phase portrait of this d.e. W510 vb?" wk: 2— (b) Classify the critical points as asymptotically stable, unstable, or semi—stifle. t. > 0. A {kin} FOL (ﬁfe {’ (AS 46 At in m , SOL V,‘ 31m LI, F1111 a} ' “as; '1 ‘3 Us 3‘ ’f) ‘3 a :7 N5 3, I: ) (JAVA (c) Roughly draw the solution curve, Without solving the d.e., for y(0) = 1/2. Midterm 1,. May 2011 MATH 201 Page 3 3. (a) Show that the differential equation
(36% + 11:) dcc + 8”” dy = 0 is not exact. ﬂ _ 3,5,; A b. i 6,5;
so 0v» 3‘0. LC: ﬂJf 12,4 qucf OWL (b) Solve this d.e. by determining an integrating factor which makes it exact. Hint:
You need to check if the quotient (My — Nx)/N depends only on 3:, or if the
quotient (Nx ~ My)/M depends only on y.) iwd‘ #3 w ,‘ c1"
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L Midterm 1, May 2011 V MATH 201 Page 4 [6] 4. (a) A population of trout in a lake is assumed to grow exponentially. Suppose that at
this moment there are 1000 trout in the lake. Also, suppose that next year 1200
trout will be in the lake. Find P(t) using the appropriate model. ‘ [if _ kt
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It k—iZ. Lajmhﬂz) (b) Suppose that 100 trout will be harvested from the lake at the end of each year.
That is, suppose that one year from now, 1100 trout Will remain after harvesting. Find P(t) using these two facts and the appropriate model. Hint: Assume that
k, the growth constant, is the same as in part (a). JP who!) : mm) 9"” o"t #
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Phh HM a“? C : Moo — 177M) Midterm 1, May 2011 MATH 201 Page 5 [5] 5. (a) Show that the d.e. ' A X y(1ny —1na: — 1)d:z: +03dy = 0 is a homogeneous, and states its degree of homogeneity. {X I
(AM— "szu/ to 5h5w' “MN/8‘ firik);(n6)x f. U/bl(l"lbb) i Nhhm) ’( ) MSV8L\A&)'\) .13 ﬂ(‘11,40)c ta (1% 61:9“ 5 x
N s w Nchﬂkby ,, h ilk NINE) 56 W‘ Viiﬂ‘b‘d“, 0”" (b) Solve this d.e. by knowing that it is a ﬁrst order homogeneous equation for an
initial condition y(6) = 1. ,Ejjl/ r Jr; at"?
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\n V\ l" ) A“? C  ’i Midterm 1, May 2011 MATH 201 Page 6 [6] 6. Solve the IVP d
£+xy=xy3ﬁ y(0) = _2
and write the solution explicitly.
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