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Unformatted text preview: MTH 205 I Review problems / Exam 1 I Summer 08 Q1. Consider the differential equation: y'= Z + J; x
a) Determine whether the existence theorem guarantees that the differential equation
possesses a unique solution through the point i) (0,0) ? ii) (1,3) ? Give reasons. b) Solve the differential equation, and give the solution intervals.
c) Find the solution that satisﬁes the initial conditionya) = 3 , if any. Q2. Classify each of the following differential equations as, separable, exact, linear,
homogeneous, or Bernoulli and solve it. a) 2ml+y2=2x2, b) yi=xy+J_’ c) y,=‘5x3+4y
8y —4x
. . ' . x2+2“
d)(1+x)y+2y=3x+3, e) y: y+x, 1) y: y
1 2+ye"
'=——*, n '=6 +12 4 2” ‘ '=———
g”) xix—3’) )1? y xy 0y 2y—xex” Q3. Consider the differential equation y' = 5 + 3y — y2 . a) Find all critical points and the phase portrait. b) Classify each critical point as asymptotically stable, unstable, or
semistable. 0) Sketch an approximate solution curve that passes through each of
the following points: i (1,—2), ii (— 1,2), iii (0,—_4), iv (0,6). Q4. A body at a temperature of 50° F is placed outdoors where the temperature is
100° F. if after 5 minutes the temperature of the body is 60° F, use the Newton's law
of cooling to ﬁnd an expression for the temperature T of the body after t minutes. 120, 05:3 20
0, 11> 20 series in which the inductance is 20 henries and the resistance is 2 ohms. Find the current 1'0) if 1(0) = 0 Q5. An eiectromotive force— E(t) = { is applied to an LR Q6. Initially 100 milligrams of radioactive substance were present. After 6 hours the mass had decreased by 3%. If the rate of decay is proportional to the amount of
the substance present at any time, ﬁnd the amount remaining after 24 hours. 120, OStSZO Q7. An electromotive force E (t ) = is applied to an LR
0 , t > 20 series in which the inductance is 20 henries and the resistance is 2 ohms. Find the current [(1‘) if 1(0) = 0 ...
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 Spring '11
 sadik
 Differential Equations, Equations

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