Obtain the differential equation satisfied by the

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Obtain the differential equation satisfied by the family of curves defined by the equation (*) below. (b) (3 pts.) Next, write down the differential equation that the orthogonal trajectories to the family of curves defined by (*) satisfy. (c) (3 pts.) Finally, solve the differential equation of part (b) to obtain the equation(s) defining the orthogonal trajectories. (*) y e cx (a) (b) (c) The differential equation in part (b) has no constant solutions, when y is considered a function of x , and separating variables yields . Integrating, and clearing the common denominator provides us with , an equation for the orthogonal trajectories.
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MAP2302/FinalExam Page 5 of 6 ______________________________________________________________________ 6. (10 pts.) The equation has a regular singular point at x 0 = 0. Substitution of into the ODE and a half a page of algebra yields Using this information, (a) write the form of the two linearly independent solutions to the ODE given by Theorem 6.3 without obtaining the numerical values of the coefficients of the series involved, and then (b) do obtain the the numerical values of the coefficients c 1 , ... , c 5 to the first solution, y 1 ( x ), given by Theorem 6.3 when c 0 = 1. [ Keep the parts separate. ] (a) Plainly, the indicial equation is r 2 - 1 = 0, with two roots r 1 = 1 and r 2 = -1. Consequently the two linearly independent solutions provided by Theorem 6.3 look like the following: and . (b) When r = r 1 = 1, (1+ r ) 2 - 1 0. Thus, we must have c 1 = 0, and Using this recurrence and the given information, it follows that ______________________________________________________________________ 7. (10 pts.) Solve the following second order initial-value problem using only the Laplace transform machine. By taking the Laplace Transform of both sides of the differential equation, and using the initial conditions, we have By taking inverse transforms now, we quickly obtain
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MAP2302/FinalExam Page 6 of 6 ______________________________________________________________________ 8. (10 pts.) (a) Given f ( x ) = x is a nonzero solution to , obtain a second, linearly independent solution by reduction of order. (b) Use the Wronskian to prove the two solutions are linearly independent. (a) Set y = vx . Substituting y into the ODE above and simplifying the algebra results in Then, setting w = v , and a little additional algebra provides us with the linear, 1st order, homogeneous equation An integrating factor for this ODE is μ = x 2 /( x 2 - 1). Solving the ODE, an
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