test1 - x 2 dx dt = 2 t +1. Give your answer with x as an...

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DEPARTMENT OF MATHEMATICS NORTH CAROLINA STATE UNIVERSITY MATH 341 TEST 1 Spring 2009 NAME: ROW and SEAT: On all problems, you must show your work to receive credit. Calculators may be used only to do arithmetic and evaluate the numerical value of expres- sions (for example, e 2 or sin ( . 7)). They may not be used for differentiation, integration, algebra, solving differential equations, graphing, etc. 1. (8 points) Use the differential equation t 2 d 2 x dt 2 - ( t 2 + t +1) dx dt = ( t +1) x to answer the following questions. (a) What is the order of the differential equation? (b) Is the differential equation linear? (c) What is the dependent variable? (d) What is the independent variable? 2. Consider the differential equation (3 x 2 y +5 x 4 - e y ) dx +( x 3 - x e y ) dy = 0. (a) (5 points) Show the differential equation is exact. (b) (15 points) Solve the initial value problem (3 x 2 y + 5 x 4 - e y ) dx + ( x 3 - x e y ) dy = 0; y (1) = 0 3. (18 points) Find a solution to the differential equation 3
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Unformatted text preview: x 2 dx dt = 2 t +1. Give your answer with x as an explicit function of t , if possible. 4. (18 points) Solve the dierential equation dy dx + 2 x x 2 + 1 y = 4 x . Give your answer with y as an explicit function of x , if possible. TWO MORE PROBLEMS ON BACK 2 5. A 500 gallon tank initially contains 100 gallons of water into which 5 pounds of salt has been dissolved. Pure water ows into the tank at 5 gallons per minute and the well-mixed salt water ows out of the tank at 1 gallon per minute. Let x ( t ) represents the pounds of salt in the tank at time t . (a) (3 points) After how many minutes will the tank overow? (b) (10 points) Give an equation for dx dt but do not solve the equation. (c) (3 points) State the initial condition, x (0). 6. (20 points) Solve the initial value problem 2 y 00 + 3 y-5 y = 0; y (0) = 10; y (0) = 3...
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test1 - x 2 dx dt = 2 t +1. Give your answer with x as an...

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