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Unformatted text preview: Test 1A Fall 06 Key 1. 4 The differential equation will be 1st order meaning therefore we will find 2 Solve for C either using y or . I would recommend to solve for C by using 2 Now plug in C into the y equation 2 4 4 multiply by 2 2 8 2. 2 First you have to divide the entire equation by x, because you want the coefficient of to be one 2 cos Integrate whatever is in front y and raise it to the exponential 2ln Multiply the whole equation by what you get 2 cos The left side of the equation becomes a product rule cos ; Now integrate both to solve for y cos sin sin 2 3. 3 5 2, 5 3 10 2 0 10 0 . Solve for r The solution for this 2nd order equation will be in the form of 0 General solution but they want fundamental set of solution , 4. 10 25 0 . Solve for r, same as the The solution for this 2nd order equation will be in the form of last problem 10 5 25 5 0 0 5. 3 cos 4 The only time we get cosine for our solution is when r equal imaginary numbers The coefficient of 3 doesn’t matter 2 2 2 4 4 4 4 20 20 4 4 4 0 0 16 5, 5 since r is the same we have to multiply one of the solutions by x 6. 2 This is a separable equation because the denominator you can factor a y and 2 2 2 ln e 2 2 4 7. 4 13
nd 2 4 1 4 2 ln · 4 2 0 . Solve for r √ Integrate both sides by u substitution 4 both sides to get ride of the ln The solution for this 2 order equation will be in the form of 4 4 4 4 4 2 2 √16 2 √ 36 2 6 3 Recall: cos 3 sin 3 cos 13 0 this cannot be factor so we use quadratic equation 4 1 13 4 21 52 sin 8. The solution is in the form ; 7 7 7 8 5 5,3 ; 15 3 0 0 15 ; 0 plug it in to the differential equation 15 15 0 0 0 a) Find two independent solutions Find the wronskian to show that they are independent 3 5 3 b) General solution c) Given the initial conditions 1 5 1 1 1 0 3 5 2 3 2 1 3 ; 2 3 2 1 5 2 5 2 3 2 5 2 1; 1 0; 0 5 3 3 1 1 1 2 1 3 1 1; 1 0 5 2 0 therefore independent Multiply eqn 1 by ‐3 and added to eqn 2 9. 2 a) Find the differential equation The differential equation will be 1st order meaning therefore we will find 3 4 Solve for C either using y or . I would recommend to solve for C by using 3 4 Now plug in C into the y equation 3 4 2 2 This is the differential equation you can simplify it but wouldn’t recommend it. b) Find the differential equation for the family of orthogonal trajectories To find orthogonal trajectories we will replace for 3 4 3 4 c) Find the family of orthogonal trajectories To find family we are going to solve the differential equation from part b 3 4
3 4 from the equation in part a 1 2 2 2 We will solve this differential equation by separating the x’s and y’s 2 2 2 2
2 3 3 4 3 4 3 4 2
2 3 2 3 4 3 4 2
2 2 1 2 3 8 You can try to simplify but I wouldn’t recommend it. 10. 4 4 a) Find a fundamental set of solutions to the reduced equation 4 4 2 2, 2 General solution , b) Find a solution of the nonhomogeneous equation This means to solve for z where Define a , Recall: . I defined them as , 2 , , 2 . 0 Fundamental set 4 2 4 0 0 0 Reduce equation 2 2 Therefore independent solutions 1 1 ln ln c) Give the general solution ofthe nonhomogeneous equation Here you will put the solutions from part a and part b together ln Notice that you can combine ln 1 ...
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 Fall '06
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