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 = δ ( t ) , y (0) = 0 , y ′ (0) = 1 . L{ y ′′ + 4 y } = L{ δ ( t ) } ; L{ y ′′ } + 4 L{ y } = L{ δ ( t ) } , L{ y ′′ } = s 2 L{ y }− sy (0) − y ′ (0); s 2 L{ y }− sy (0) − y ′ (0) + 4 L{ y } = L{ δ ( t ) } , y (0) = 0 , y ′ (0) = 1 , L{ δ } = 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 = λ ± iμ , λ > 0 implies an unstable spiral point . (d) r = ± 2 i . r = ± iμ 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, − π ≤ x ≤ π, c ( x + 2 π ) = c ( x ); d ( x ) = 1 + sin x, − π ≤ x ≤ π, d ( x + 2 π ) = d ( x ); e ( x ) = x − x 3 , −...
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C251final(sp07) - Math 251 Spring 2007 final exam solutions...

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