EM-Ex1s-Sp09 - Exam 1 Review Show that y = y (x) is/is not...

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Exam 1 Review Show that y = y ( x ) is/is not a solution of the differential equation F ( x,y,y ± ,...y ( n ) )=0 . Examples: 1. x 2 y ±± - 3 xy ± +4 y =0 ; y 1 ( x )= x 2 ,y 2 ( x x 2 ln x 2. d 3 y dx 3 + dy dx = e x ; y ( x )=1+sin x + 1 2 e x ,z ( x ) = 2 cos x + 1 2 e x . 3. xy ±± + y ± ; y 1 ( x ) = ln (1 /x ) 2 ( x x 2 . 4. ( x +1) y ±± + xy ± - y =( x 2 ; y ( x e - x + x 2 +1 ( x x 2 +1. 5. y ± + y = y 2 ; y = 1 Ce x . Find the differential equation for an n -parameter family of curves. Examples: 1. y 2 = Cx 3 - 2. 2. y = C 1 x 3 + C 2 x . 3. y = C 1 e - 2 x + C 2 xe - 2 x . 4. y = C 1 x 3 + C 2 First order linear equations; ±nd general solution, solve an initial-value problem. Examples: 1. Find the general solution of x 2 dy - 2 xy dx = x 4 cos 2 xdx . 2. Find the general solution of (1 + x 2 ) y ± +1+2 . 3. Find the general solution of xy ± - y =2 x ln x . 4. Find the solution of the initial-value problem xy ± +3 y = e x x (1) = 2. Separable equations; ±nd general solution, solve an initial-value problem. Examples: 1. Find the general solution of yy ± + xy 2 - x . 2. Find the general solution of y ± = xe x + y . 3. Find the general solution of ± = xy 2 - x - y 2 1
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4. Find the general solution of ln x dy dx = y x . 5. Find the solution of the initial-value problem y ± = x 2 y - y y +1 ,y (3) = 1. Applications Examples: 1. Given the family of curves y = Ce x . Find the family of orthogonal trajectories.
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This note was uploaded on 01/30/2011 for the course MATH 3321 taught by Professor Morgan during the Spring '08 term at University of Houston.

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EM-Ex1s-Sp09 - Exam 1 Review Show that y = y (x) is/is not...

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