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A portion of an exam from a quantitative methods in hydrology course at new mexico tech, focusing on solving ordinary differential equations (odes) through various methods. It includes short answer questions related to finding particular solutions, verifying solutions, testing for exactness, and identifying linearity.
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Quantitative Methods in Hydrology Part A New Mexico Tech October 31, 2007
-1 0 1 2 3 4 x
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y
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PART A Short Answers (60 pts out of 100pts)
(1) Solve the ODE by integration y’ = e -3x
(2) State the order of the ODE. Verify that the given function is a solution. Show your work.
y’’ + 2 y’ + 10 y = 0, y= 4 e -x^ sin(3 x )
(3). Verify that the given y is a solution of the ODE. c is a constant. Determine from y the particular solution satisfying the given IC. Sketch the graph of the solution.
y’ = 0.5 y , y = c e 0.5 x , y (2) = 2
(4) Does the ODE y’^2 = -1 have a real solution?
(5) Consider the ( x,y ) direction field graph to the right. In the field draw curves that approximate the solutions that pass through the points (1,1), and (-1.5, 0).
(7) Find a general solution of the following. Show the steps of the derivation. Show your check of the solution by substitution.
y’ = 2 sec 2y
For part A there are 14 problems, each worth five points. Solve 12 correctly to get full credit of 60 points. If you have difficulty with a problem, can recognize the difficulty, and can explain how you know you are having difficulty, you may get partial credit.
For part B there are 5 problems at five points each. These are essay questions. For part C there is one problem worth 15 points, involving programming issues. It will be given out separately. Exercise your judgment to select the problems to solve. Some problems take a few minutes, others take longer. The credit given does not necessarily correspond to the time the problem takes.
Quantitative Methods in Hydrology Part A New Mexico Tech October 31, 2007
(7) Find the particular solution. Show the steps of the derivation, beginning with the general solution which you should find using SOV.
y y’ + 4 x = 0, y (0) = 3
(8) Recall the definition of an exact ODE. Test the following equation for exactness. If exact solve. If not, use the given integrating factor or find the solution by inspection.
ex^ (cos x dx – sin y dy ) = 0
(9) Is the given equation linear? Is it homogeneous or non-homogeneous? Rewrite it in standard form. Then find its general solution. If linear, consider using an integrating factor. If non-linear try to reduce to linear form, first.
y’ = 4 y + x
(10) Is the given equation linear? Is it homogeneous or non-homogeneous? Rewrite it in standard form. Then find its general solution. If linear, consider using an integrating factor. If non-linear try to reduce to linear form, first.
y’ + k y = e^2 kx
Quantitative Methods in Hydrology Part B New Mexico Tech October 3, 2007
PART B Concise essay answers. (25 pts) Give your essay answers in proper English, as if you were explaining to a lay person.
2.What is the difference between a general solution of an ODE and a particular solution? Consider in the context of 1 st^ order ODEs.