Math 261 practice test4

1) y’’’ – y’ = 0, y (0) = 3, y’ (0) = 0, y’’ (0) = 2

2) Assume that y’’ – 5y’= 0 has a power series solution ∑ 𝑎𝑛∞𝑛=0 𝑥𝑛 .

Find the recurrence relation for 𝑎𝑛, 𝑛 = 0, 1, 2, …

Problems 2, 3, and 4: use either the table 6.21 on page 321 or my handout to find the Laplace Transform

and the inverse of the Laplace Transform.

3) Find the Laplace Transform for:

a) f (t) =3 e – 4t + 5e4t + 7, b) f (t) = 2t e3t + 5e3t, c) f(t) = 3 e – 2t cos(√5t) – e – 2t sin(√5t)

d) 3cos(5t) – 4sin(5t), e) 5e2tcos(t) – 3e2t sin(t)

4) Find the Inverse of the Laplace Transform for:

a) F(s) = 2/s+ 3/(s – 1) + 2/ (s – 1 )2 b) F(s) = (5s + 1)/[(s+ 2)( s – 1 )], c) F(s) = (s + 5)/(s2 + 4),

d) Y(s) = (s + 3)/(s2 – 2s + 5), e) Y(s) = (2s + 1)/[s(s + 2)]

5) Use the Laplace Transform to solve the given Initial Value Problems:

a) y’’ – 5y’ + 6y = 0, y(0) =3, y’(0) =8

b) y’’ + y’ = 0, y(0) =1, y’(0) =1

c) y’’ + 4y = 0, y(0) = 2, y’(0) = 6

6) Transform the given differential equation into a system of the first order differential equations:

a) u’’’ – 2u’’ + u’ – u = 0, b) y’’ – 3y’ + 2y = 0

7) Verify that the given vector satisfies the given system of differential equations

( 1 − 13 5

) 𝑋 = 𝑋′ if: a) 𝑋(1) = ( 1

– 1) 𝑒2𝑡, b) 𝑋(2) = (

1−3

) 𝑒4𝑡

Then solve the initial value problem: 𝑋 (00

) = (21

), assume 𝑋 = 𝑐1𝑋(1) + 𝑐2 𝑋(2)

8) Find all eigenvectors and eigenvalues of the matrix a) 𝐴 = ( 1 22 1

) b) 𝐴 = ( 2 4 3 3

)

Good Luck!

FORMULA SHEET

f(t) = 𝐿−1{𝐹(𝑠)} 𝐹(𝑠) = 𝐿(𝑓(𝑡) f(t) =𝐿−1{𝐹(𝑠)} 𝐹(𝑠) = 𝐿(𝑓(𝑡)

1 1

𝑠 , 𝑠 > 0 𝑒𝑎𝑡

1

𝑠−𝑎 , 𝑠 > 𝑎

a 𝑎

𝑠 , 𝑠 > 0 𝑡𝑒𝑎𝑡

1

( 𝑠−𝑎)2 , 𝑠 > 𝑎

cos(𝑏𝑡) 𝑠

𝑠2+𝑏2 , 𝑠 > 𝑎 𝑠𝑖𝑛(𝑏𝑡) 𝑏

𝑠2+𝑏2 , 𝑠 > 𝑎

𝑒𝑎𝑡𝑐𝑜𝑠(𝑏𝑡) 𝑠−𝑎

(𝑠−𝑎)2+𝑏2 , 𝑠 > 𝑎 𝑒𝑎𝑡𝑠𝑖𝑛(𝑏𝑡) 𝑏

(𝑠−𝑎)2+𝑏2 , 𝑠 > 𝑎

Solving Initial Value Problem y’’ + by’ + cy = 0, y(0) , y’(0) if y(t) is the solution, then

𝑌(𝑠) = 𝑠𝑦(0) + 𝑦′(0) + 𝑏𝑦(0)

𝑠2 + 𝑏𝑠 + 𝑐

Partial Fractions: 1) ∆ > 0 𝛼𝑠+ 𝛽

(𝑠−𝑠1 )(𝑠−𝑠2) =

𝐴

𝑠−𝑠1+

𝐵

𝑠− 𝑠2

2) ∆ = 0: 𝛼𝑠+ 𝛽

(𝑠−𝑠1 ) 2 =

𝐴

𝑠−𝑠1+

𝐵

(𝑠−𝑠1)2

3) ∆ < 0: complete the square 𝑠2 + 𝑏𝑠 + 𝑐 = (𝑠 + 𝐴)2 + 𝐵, 𝐵 > 0

System of Linear D.E. of the first order𝑋′ = (𝑎11 𝑎12

𝑎21 𝑎22) 𝑋, 𝑋 = (

𝑣1

𝑣2) 𝑒𝜆𝑡

To find the eigenvalues𝜆𝑖: set |𝐴 − 𝜆𝐼2| = 𝜆2 − (𝑎11 + 𝑎22)λ + 𝑎11𝑎22 − 𝑎12𝑎21 = 0,

To find an eigenvector (𝑣1

𝑣2) corresponding to the eigenvalue 𝜆𝑖 solve

(𝑎11 − 𝜆 𝑎12

𝑎21 𝑎22 − 𝜆) (

𝑣1

𝑣2) = (

00

). Assume 𝜆1 ≠ 𝜆2

Ludmila Bobek-Smith