Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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- Exercise 1. A student is asked to prove If m is even and n is odd, then m² +n² is odd. The student provides the following proof: Proof. If m is even and n is odd, then m² + n² is odd. Since m is even, m = = 2k for any integer k. Since n is odd, n = 2k + 1. Note that m² + n² = (2k)² + (2k + 1)² = 8k² + 4k +1 So m² + n² is odd because it is an even number plus 1. (a) Identify the errors in the proof above. There are multiple errors. Explain each error. Your explanation should be written to the student who made the error and should try to help the student understand why what they wrote is incorrect. (b) In addition to errors, provide suggestions to improve the student's proof writing. That is, explain what parts of the proof are unclear or unjustified. Explain how such parts of the proof prevent some students from fully following the ment made. (c) Finally, provide a correct proof of the theorem above.arrow_forwardAnswer the following questions and show all of your work. (a) Sketch the following function where u_a(t)=u(t-a), the unit step function. f(t)=(t-1)(u_1(t)-u_2(t)) +u_2(t). (b) Can the following first order differential equation be solved using the Laplace Transform Method? y'+y=6e^{t^2}, y(0)=3. (c) Can the following first order differential equation be solved using the Laplace Transform Method? y'+3y=2e^{-t},y(0)=3. If not, state clearly why not and if it can be solved, find the solution and show all of your steps.arrow_forwardCan someone solve this ODE problem using the Bernoulli equation, thank you!arrow_forward
- Consider the following initial-value problem. y' + y = t sin(t), y(0) = 0 Take the Laplace transform of the differential equation and solve for L{y}. (Write your answer as a function of s.) L{y} = Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms as needed. y(t) =arrow_forwardConsider the initial value problem + 2y = 12t, y(0) = 6. a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b) below). help (formulas) b. Solve your equation for Y(s). Y(s) = L{y(t)} = c. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t). y(t) =arrow_forwardL{sin(T – «(T – } 3n. 2arrow_forward
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