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Quantitative Methods in Hydrology Part A New Mexico Tech October 31, 2007
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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.
- Why are some ODEs called homogeneous and others called non-homogeneous? What is the difference?
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.
- What does the truncation error of the backward Euler Method depend on? How do you improve accuracy (reduce truncation error) with this method?
- Which Euler schemes are conditionally stable? For the most restricted scheme (most conditionally stable), what is restricted (constrained or limited) in order to ensure stability and why?
- How does a Runge-Kutta (RK) method differ from any of the Euler Methods? Consider both the algorithms and their truncation error. How can you improve (reduce) the truncation error of a RK method?
