C251final(sp07)

C251final(sp07) - Math 251 Spring 2007 final exam solutions...

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Unformatted text preview: Math 251 Spring 2007 final exam solutions Pbm. 1. Which of the following equations below is a second order, linear, nonhomogeneous, ordinary differ- ential equation? (a) ( y ) 2 ty = e t , first order, nonlinear (b) y + 2 y = 1 y , second order, nonlinear (c) t 2 y + 2 ty + 4 y = 1 (d) y + 3 y + 2 y = 0, second order, linear, homogeneouos (e) 2 y 3 y = t 1, third order, linear, nonhomogeneous Pbm. 2. Find the solution of the initial value problem y + 4 y = ( t ) , y (0) = 0 , y (0) = 1 . L{ y + 4 y } = L{ ( t ) } ; L{ y } + 4 L{ y } = L{ ( t ) } , L{ y } = s 2 L{ y } sy (0) y (0); s 2 L{ y } sy (0) y (0) + 4 L{ y } = L{ ( t ) } , y (0) = 0 , y (0) = 1 , L{ } = 1; ( s 2 + 4) L{ y } 1 = 1; L{ y } = 2 s 2 + 4 ; This is the Laplace transform of y = sin(2 t ) . Pbm. 3. Consider the initial value problem t 2 y + 2 y = 1 t 2 16 , y ( 2) = 1 . What is the largest interval on which a unique solution to this initial value problem is certain to exist? In order to be able to apply the corresponding existance and uniqueness theorem , one needs to write the equaion in the form y + p ( t ) y = g ( t ), i.e. divide the equation by t 2 : y + 2 t 2 y = 1 t 2 ( t 2 16) . In this form p ( t ) = 2 t 2 , which is not defined for t = 0, and g ( t ) = 1 t 2 ( t 2 16) , which is not defined for t = 4, t = 0, and t = 4. So the maximal intervals, on which a solution is certain to exist are ( , 4), ( 4 , 0) (0 , 4), and (4 , ). The initial condition y ( 2) = 1 is given at t = 2, which is in the interval ( 4 , 0) . 1 Pbm. 4. What is the inverse Laplace transform of F ( s ) = e 2 s 1 ( s 1) 2 ? L 1 { 1 ( s 1) 2 } = e t L 1 { 1 s 2 } = e t t ; L 1 { e 2 s 1 ( s 1) 2 } = u 2 ( t ) e t 2 ( t 2) . Pbm. 5. Consider systems of the form vector x = Avectorx . For each pair of eigenvalues for the matrix A , listed below, state the type and stability of the critical point at (0 , 0). (a) r = 3 , 2. r 1 negationslash = r 2 < 0 implies an asymptotically stable node . (b) r = 3 , 2. r 1 < < r 2 implies a saddle point , (automatically) unstable . (c) r = 1 5. r = i , > 0 implies an unstable spiral point . (d) r = 2 i . r = i implies a center , (automatically) stable, but not asymptotically . (e) r = 3 , 5. r 1 negationslash = r 2 > 0 implies an unstable node . Pbm. 6. Consider the following periodic functions: a ( x ) = x, 2 < x < 2 , a ( x + 4) = a ( x ); b ( x ) = x 2 , 1 x 1 , b ( x + 2) = b ( x ); c ( x ) = 1 + cos x, x , c ( x + 2 ) = c ( x ); d ( x ) = 1 + sin x, x , d ( x + 2 ) = d ( x ); e ( x ) = x x 3 ,...
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C251final(sp07) - Math 251 Spring 2007 final exam solutions...

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