MATH 2300 Exam 2 (Spring 2007) & solutions

# MATH 2300 Exam 2 (Spring 2007) & solutions - MATH...

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Unformatted text preview: MATH 2300 (Calculus 2), Spring 2007 — 2nd Test — SOLUTIONS 1 MATH 2300 (Calculus 2) Spring 2007 2nd midterm exam — SOLUTIONS 1. [10 pts] Solve d y d x + 2 xy = x by the method of integrating factors. SOLUTION From the general form of the equation d y d x + p ( x ) y = q ( x ) we obtain p ( x ) = 2 x and q ( x ) = x . Therefore μ ( x ) = exp p ( x ) d x = exp 2 x d x = e x 2 and d y d x + 2 xy = x = ⇒ d d x e x 2 y ( x ) = e x 2 x = ⇒ d d x e x 2 y ( x ) d x = e x 2 x d x = ⇒ e x 2 y ( x ) = C + 1 2 e x 2 = ⇒ y ( x ) = Ce- x 2 + 1 2 2. [10 pts] Solve d y d x = 2 ( 1 + y 2 ) x by the method of separation of variables. SOLUTION d y d x = 2 ( 1 + y 2 ) x = ⇒ d y 1 + y 2 = 2 x d x = ⇒ d y 1 + y 2 = 2 x d x = ⇒ arctan y = x 2 + C = ⇒ y = tan ( x 2 + C ) 3. [15 pts] At time t = 0 a tank contains 10 oz of salt dissolved in 20 gal of water. Then brine containing one oz of salt per gallon of brine is allowed to enter the tank at a rate of two gallons per minute, and the mixed solution is drained from the tank at the same rate. How much salt is in the tank at an arbitrary time t?...
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MATH 2300 Exam 2 (Spring 2007) & solutions - MATH...

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