ࡱ> 574U@ 0&bjbj "V(((((((<DDDD|<<,+++++++$,R/+(+((+> > > ((+> +> x> !r_*T((_+ /uѣD*++0,*/X/_+<<((((/(_+ > ++<<d> <<Differential Equations Final Practice Exam (Final Spring 1996 Problem 2) Find the general solution of the following differential equations. a)  EMBED Equation.DSMT4  b)  EMBED Equation.DSMT4  Solutions Integrating Factor Method:  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . Let  EMBED Equation.DSMT4   EMBED Equation.DSMT4  First find the homogeneous solution and then find the particular solution to find the general solution. Since the roots of the characteristic equation equals are the same as exponent in front of the exponential, you will need to add a factor of  EMBED Equation.DSMT4  because of the repeated root of the homogeneous solution. This problem is basically the same as Example 6 on p. 234 with exception that you should find that A=1/2.  EMBED Equation.DSMT4  (Final Spring 1996 Problem 3) Consider the differential equation  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . According to the theorem on existence and uniqueness, on what interval of x is the solution guaranteed to exist and be unique? Find the solution of the equation. On what interval of x does the solution exist? Solutions a)  EMBED Equation.DSMT4 . Check if well defined at initial value.  EMBED Equation.DSMT4 . So it is well defined. Check if  EMBED Equation.DSMT4  is well defined at the initial values  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Since both  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are both well defined at the initial values, a unique solution exists surrounding the value x=0. The only restriction is on x because sec x is defined on the interval  EMBED Equation.DSMT4 . b) Homogeneous equation  EMBED Equation.DSMT4   EMBED Equation.DSMT4  I will use the method as outlined on p.419 takes a little less work. Although you can use this method on the exam, (WARNING) I heard that no partial credit will be given if you make a mistake!!! So try the problem as professors taught section 8.2 to give you practice to do it either way. It really depends how good you are at memorizing. Solving  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 . Let  EMBED Equation.DSMT4  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4   EMBED Equation.DSMT4  General solution  EMBED Equation.DSMT4  (Final Spring 1996 Problem 4) A tank contains 10,000 liters of a solution consisting of 100 kg of salt dissolved in water. Pure rocky mountain spring water is pumped into the tank at the rate of 10 liters/second. (The mixture is kept uniform by stirring.) The mixture is pumped out of the tank at the same rate it enters. Set up the initial value problem for the amount of salt, S(t), in the tank at time t. Solve this problem for S(t). How long will it be before only 50 kg of salt remain in the tank? Solutions Salt inflow is 0 Salt outflow is  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  b)  EMBED Equation.DSMT4  Using the initial condition,  EMBED Equation.DSMT4  c)  EMBED Equation.DSMT4  (Final Spring 1996 Problem 8) A damped spring is connected to a crank, which forces the spring at a frequency of 1/sec. According to the laws of Newton and Hooke, the governing equation is  EMBED Equation.DSMT4 . Explain the physical meaning of each term in the equation. Find the general solution of the equation. If the spring is steadily cranked for 2 hours, determine the amplitude of the solution. Solutions a)  EMBED Equation.DSMT4 represents mass times acceleration  EMBED Equation.DSMT4  represents the damping force (friction) u represents the restoring force 10sin t represents external driving forces in this case the crank b)  EMBED Equation.DSMT4  Homogeneous solution:  EMBED Equation.DSMT4  Particular solution: Guess for a particular solution  EMBED Equation.DSMT4  Solving for coefficients A and B  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Thus  EMBED Equation.DSMT4  General solution:  EMBED Equation.DSMT4  At this point, the solution to part c I dont know not the same as calculating SHO where we can calculate the amplitude using the formula on p.196. I would not worry about this problem since you were not asked about this before.. Just make sure you know how to find the amplitude of SHO. At time t=0,  EMBED Equation.DSMT4  At time t=2  EMBED Equation.DSMT4  Period of the oscillation is  EMBED Equation.DSMT4  (Final Spring 1996 Problem 10) Solutions a)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Since alpha = 0 (i.e. the eigenvalue is purely imaginary), the solution is a center and is stable. b)  EMBED Equation.DSMT4  Since the eigenvalues are real distinct roots, with one eigenvalue positive and the other eigenvalue negative, the solution is unstable. Eigenvectors  EMBED Equation.DSMT4   EMBED Equation.DSMT4  c)  EMBED Equation.DSMT4  Since the eigenvalues are complex numbers and alpha=-1 is less than 0. The solution is asymptotically stable. In calculating the eigenvectors, we will utilize the fact that the eigenvectors are complex conjugates of each other since the eigenvalues are complex conjugates.  EMBED Equation.DSMT4  (Final Fall 1996 Problem 4) Solutions a)  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  Hence the equilibria (1,1) and (-1,1) b)  EMBED Equation.DSMT4  At equilibrium point (1,1),  EMBED Equation.DSMT4  Since alpha =1 is greater than 0, the solution is unstable , repelling spiral point. At equilibrium point (-1,1),  EMBED Equation.DSMT4  Since the eigenvalues are real distinct roots and one eigenvalue is positive with the other eigenvalue negative, the equilibrium point is a saddle and the solutions are unstable. (Final Fall 1998 Problem 6) For the linear system of differential equations  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Solutions a)  EMBED Equation.DSMT4  b)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  eigenvalues  EMBED Equation.DSMT4  c)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  are the eigenvectors (utilizing the fact the eigenvectors will be complex conjugates because the eigenvalues are complex numbers) Note that any multiple (where r can be any complex number) would be an acceptable solution. Since  EMBED Equation.DSMT4 , the general solution is equal to  EMBED Equation.DSMT4  e)  EMBED Equation.DSMT4   EMBED Equation.DSMT4 , EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  f)  EMBED Equation.DSMT4  means trajectories are closed loops. See p. 347 of the text of how to Figure 6.3.3 and Example 4 to draw the phase-plane. (Final Fall 1998 Problem 7) For each of the following answer TRUE or FALSE. 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