Homework 2 solutions

Homework 2 solutions - AMS 361: Applied Calculus IV by Prof...

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Unformatted text preview: AMS 361: Applied Calculus IV by Prof Y. Deng Homework 2 (Problem 2-1, Prob 16, P. 41) Find the general solutions (implicit if necessary, explicit if convenient) of the differential equations in the following problem: 1 tan Solution: This is a separable equation 1 1 (implicit) or 1 explicit (Problem 2-2, Prob 28, P. 41) Find the explicit particular solution of the initial value problem 2 4 Solution: Please refer to the solution in the "Deng Notes" (Prob 2-2, HW2 Spring 2005, p 247) (Problem 2-3, Prob 16, P. 56) Solve the following differential equation. Primes denote derivatives with respect to x. 1 Solution: Please refer to the solution in the "Deng Notes" (Prob 2-4, HW2 Fall 2005, p 276) (Problem 2-4, Prob. 18, P.56) Find the general solution of the differential equation. Primes denote derivatives with respect to x. 2 cos Solution: Please refer to the solution in the "Deng Notes" (Prob 2-4, HW2 Spring 2005, p 247) (Problem 2-5, Prob. 27, P.56) Find the general solution of the differential equation (regarding x as depending variable and y as independent). Primes denote derivatives with respect to x: 1 cos 2 cos /4 AMS 361: Applied Calculus IV by Prof Y. Deng Homework 2 1 2 1 Solution: Please refer to the solution in the "Deng Notes" (Prob 2-5, HW2 Spring 2005, p 248) Method 1: Exact Equation Method: The original DE can be written as 1 1 2 0 Thus, 1 1 Then, we can have 1 Resulting in a new equation that's exact: 1 2 1 [ 1 Now, we can introduce the following two equations 1 1 1 1 Thus, we have 1 Plugging into the second equation above, we get 2 1 Thus, we get 1 or Finally, 1 Thus, the solution is , 1 1 Method 2: Change the equation to 1 2 1 as dependent variable and as independent: , 2 2 2 2 0 1 1 2 constant constant another constant and it becomes 1st order linear DE with - 2 ...
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