MTH 205-Practice problems for Test I-Summer 09

MTH 205-Practice problems for Test I-Summer 09 - MTH 205 I...

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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 semi-stable. 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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MTH 205-Practice problems for Test I-Summer 09 - MTH 205 I...

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