ASSIGNMENT
Problem 1 - In this problem, you will work with the differential equation x2y" + (3x2 + 2x)y' - 2y = 0.
1. There is one singular point for this equation. Find it and classify it as either regular or irregular.
Type or write "x = _____ is a(n) ______singular point.", where you fill in the first blank with the singular point value and the second blank with either "regular" or "irregular".
2. Find the indicial equation and the two indicial roots you obtain by beginning the method of Frobenius on the differential equation for this problem.
Calculate r1 - r2 =, where r1 is the larger indicial root and identify which of the three Method of Frobenius cases applies to this equation.
a. Type or write "The indicial equation is __________.", where you fill in the blank with the equation.
b. Type or write "The two indicial roots are r1 = _______ and r2 = _______ .", where you fill in the indicial root values, with r1 as the larger root.
c. Type or write "r1 - r2 = _____", where you fill in the blank with the answer.
d. Type or write "This equation is a method of Frobenius case ____., where you fill in the blank with "I", "II", or "III".
3. Using the smaller indicial root, find a recurrence relation that can be used to find the coefficients of the series solution.
Type or write "For r = __, a recurrence relation for the coefficients is _____________________, starting with k = ___.", where you fill in the first blank with the smaller indicial root value, the second blank with the coefficient relation equation, and the third blank with the smallest value of k that the recurrence equation is used with.
4. If you were to work through the problem, you would find that the general solution for this problem would be generated only by using the smaller indicial root.
Regardless of your results in paragraphs 2 and 3 above, assume that an indicial root of r = -2 produces the coefficients for the first seven terms of the general solution of x2y" + (3x2 + 2x)y' - 2y = 0 given below, using the arbitrary constants c0 and c3.
The coefficients of the Frobenius series are:
c0 is the arbitrary constant for the first term (n = 0)
c1 = -3c0(n = 1)
c2 = 9/2c0(n = 2)
c3 is the arbitrary constant for the fourth term(n = 3)
c4 = -(3/4)c3(n = 4)
c5 = 9/20c3(n = 5)
c6 = -(9/40)c3(n = 6)
Express the series solution to the problem by using the first seven terms of the general solution.
Type or write "y = _______________________.", where you list the first seven terms of the solution in the blank. Use c0 and c3 as the arbitrary constants of your solution.
5. Use a computer program or computing algebra system (CAS) calculator to find an analytic (non-series) solution.
Type or write "y = ____________________.", where you fill in the analytic solution in the blank.
Use c1 and c2 as the arbitrary constants of your solution.
Suggested Internet site: https://www.wolframalpha.com/examples/Calculus.html
NOTES:
a. If you enter the differential equation and click the "=" button on the right of Wolfram Alpha's command line, the program will solve the problem!
b. You do NOT need to use an asterisk to represent multiplication operations; however, you may do so if you wish. For example, to enter the formula "2x", you can type "2x".
c. You must use a carat "^" to raise an expression to a power. For example, to enter the formula "2x", you type "2^x".
d. You can use "e" to represent the natural base e. For example, to enter the formula "e2x", you can type "e^2x".
e. You can use either differential or prime notation to enter a derivative expression. For example, you can express the derivative of y with respect to x as either "dy/dx" or "y'".
6. Show that the solutions you obtained in paragraphs 4 and 5 are equivalent.
Type or write an explanation that shows that the solutions are equivalent.
HINT: The general solutions you gave in paragraphs 4 and 5 should each have two linearly independent solutions (the two parts that are multiplied by different arbitrary constants). The c0 part of the paragraph 4 solution should have only three terms. The c3 part of the paragraph 4 solution has an infinite number of terms. One part of the paragraph 5 solution should be an algebraic expression: it does not contain e. The other part of the paragraph 5 solution should contain e: it is non-algebraic. To begin, take the algebraic part of the paragraph 5 solution and show that it is equivalent to the c0 part of the paragraph 4 solution. Next, take the other, non-algebraic part of the paragraph 5 solution and find a series expansion for it. You can use the "series" command in Wolfram Alpha to do this, if you wish. Compare the result to the c3 part of the paragraph 4 solution. Are they the same? Manipulate the two general solution forms of paragraphs 4 and 5 to show that they are equivalent. Find equations relating c0 and c3 to c1 and c2 that show that the solutions are equivalent.
Introduction to Problems 2 and 3:
An Army cannon is designed so that when a shell is fired from it, the carriage that the gun tube assembly sits on remains stationary, while the gun tube assembly slides in recoil, as shown in the diagram below.
When a shell is fired, two components between the gun tube assembly and the carriage absorb the recoil: a damping mechanism and a recoil spring. The recoil spring has the additional function of pushing the gun tube assembly back to the starting position, so that another shell can be loaded and fired. The gun tube assembly has a mass of 1,600 kilograms. The recoil spring has a spring constant of 20,000 Newton's per meter. The damping mechanism exerts a force, in Newton's, numerically equal to 9,500 times the instantaneous velocity, in meters per second, of the gun tube assembly.
As you saw in Assignment, the form of a differential equation that models this problem is given by 1600x" + 9500x' + 20000x = f(t), where f(t) is a given forcing function.
Problem 2 -
For this problem, assume that when the cannon is fired, the gun tube assembly begins in the equilibrium position, the initial instantaneous velocity of the gun tube assembly is zero, and the gas pressure force from the firing of the shell pushes the gun tube assembly to the rear (to the right in the diagram) with a force of 40,500 Newton's for 1/5 of a second, the length of time it takes for the shell's powder to completely burn.
After 1/5 of a second, assume that the gas pressure force is zero.
1. Model this system as a single initial-value differential equation problem using the unit step function, u(t - a), to express the gas pressure force, where x is the displacement of the gun tube assembly from the starting position (x = 0),t is time in seconds, and a is the time in seconds at which the gas pressure force becomes zero.
Type or write "The initial-value problem is", followed by your answer.
2. Take the Laplace transform of the problem you formulated in paragraph 1.
Type or write "The Laplace transform equation of the problem is", followed by your answer.
3. Solve and simplify the Laplace transform equation you obtained in paragraph 2 to find the Laplace transform of x, X(s).
Type or write "The Laplace transform of x is", followed by your simplified answer.
4. Take the inverse Laplace transform of X(s) that you found in paragraph 3 to find the particular solution that gives the displacement of the gun tube assembly with respect to time. Your answer must contain a unit step function!!
Type or write "The particular solution is", followed by your answer.
NOTES:
a. You can find the partial fraction expansion of an expression, if it exists, by typing "partial fraction", followed by the expression. For example, type "partial fraction 3/(4s^2 - 3s -1)", and click the "=" button.
If it produces an expression using "seconds" as units, click on the highlighted "a variable" in the line below the command box.
b. You can find the inverseLaplace transform of an expression, if it exists, by typing "inverse Laplace transform", followed by the expression. This command takes a function of s and produces the inverse Laplace transform function of t.
For example, type "inverse Laplace transform 1/(1 + s)", and click the "=" button.
c. WolframAlpha uses "θ" to represent the unit step function (your textbook uses "U").
d. You do NOT need to use an asterisk to represent multiplication operations; however, you may do so if you wish. For example, to enter the formula "2x", you can type "2x".
e. You must use a carat "^" to raise an expression to a power. For example, to enter the formula "2x", you type "2^x".
f. You can use "e" to represent the natural base e. For example, to enter the formula "e2x", you can type "e^2x".
g. You can use either differential or prime notation to enter a derivative expression. For example, you can express the derivative of y with respect to x as either "dy/dx" or "y'".
h. If you enter the expression for an initial-value differential equation problem, using a comma to separate the equation and the condition(s), and click the "=" button on the right of WolframAlpha's command line, the program will solve the problem!
5. Plot the particular solution you found in paragraph 4 for the first three seconds of motion.
Problem 3 -
For this problem, assume that when the cannon is fired, the gun tube assembly begins in the equilibrium position, the initial instantaneous velocity of the gun tube assembly is zero, and the gas pressure from the firing of the shell exerts an instantaneous impulse force of 8,100Newtons on the gun tube assembly to the rear (to the right in the diagram).
1. Model this system as an initial-value differential equation problem using the Dirac delta function to express the gas pressure force, where x is the displacement of the gun tube assembly from the starting position (x = 0) and t is time in seconds.
Type or write "The initial-value problem is", followed by your answer.
2. Take the Laplace transform of the problem you formulated in paragraph 1.
Type or write "The Laplace transform equation of the problem is", followed by your answer.
3. Solve and simplify the Laplace transform equation you obtained in paragraph 2 to find the Laplace transform of x, X(s).
Type or write "The Laplace transform of x is", followed by your simplified answer.
4. Take the inverse Laplace transform of X(s) that you found in paragraph 3 to find the particular solution that gives the displacement of the gun tube assembly with respect to time.
Type or write "The particular solution is", followed by your answer.
5. Plot the particular solution you found in paragraph 4 for the first three seconds of motion.
Insert a copy of the graph into your assignment submission.
Make sure that you scale the axes so that you get a clear picture of the displacement curve.
6. At what time does the gun tube assembly first return to the starting position? Solve an equation to find the answer (do not just approximate the answer from the graph you produced in paragraph 5).
a. Type or write "The equation I solved to find the answer was", followed by your answer.
b. Type or write "The assembly first returns to the starting position after _____ seconds.", where you fill in the blank with the answer. The answer may be an exact answer or a decimal approximation accurate to 3 decimal places.
7. Use the derivative of the particular solution you found in paragraph 5 to find the time that the assembly reaches its maximum displacement from the starting position. Find the maximum displacement.
a. Type or write "The derivative of the particular solution is", followed by your answer.
b. Type or write "The assembly reaches its maximum displacement after _____ seconds.", where you fill in the blank with the answer. Express the answer as decimal approximation accurate to 3 decimal places.
c. Type or write "The maximum displacement is _____ meters.", where you fill in the blank with the answer. Express the answer as decimal approximation accurate to 3 decimal places. Make sure that the sign of your answer reflects the correct direction of displacement.