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EM-Ex1s-Sp09

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

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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 x y + 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 = 0; y 1 ( x ) = ln (1 /x ) , y 2 ( x ) = x 2 . 4. ( x + 1) y + xy - y = ( x + 1) 2 ; y ( x ) = e - x + x 2 + 1 , z ( x ) = x 2 + 1. 5. y + y = y 2 ; y = 1 Ce x + 1 . 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; find general solution, solve an initial-value problem. Examples: 1. Find the general solution of x 2 dy - 2 xy dx = x 4 cos 2 x dx . 2. Find the general solution of (1 + x 2 ) y + 1 + 2 x y = 0. 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 , y (1) = 2. Separable equations; find general solution, solve an initial-value problem. Examples: 1. Find the general solution of yy + xy 2 - x = 0. 2. Find the general solution of y = xe x + y . 3. Find the general solution of yy = xy 2 - x - y 2 + 1. 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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EM-Ex1s-Sp09 - Exam 1 Review Show that y = y(x is/is not a...

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