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Exam 1 Review - 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 Identify each of the following first order differential equations. 1. x (1 + y 2 ) + y (1 + x 2 ) y = 0. 2. ( xy + y ) y = x - xy . 3. xy 2 dy dx = x 3 e y/x - x 2 y 4. y = - 3 y x + x 4 y 1 / 3 . 5. (3 x 2 + 1) y - 2 xy = 6 x . 6. x (1 - y ) + y (1 + x 2 ) dy dx = 0. 7. xy = x 2 y + y 2 ln x . 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 . 1
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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.
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