Question 1

The inverse Laplace transform of $F ( s ) = 3 / s ^ { 2 }$ is

Accepted Answer

A)  $t ^ { 2 } / 2$ 
B)  $t ^ { 2 } / 3$ 
C)  $t ^ { 3 } / 3$ 
D)  $t / 3$ 
E)  $3 t$ 
A)  $t ^ { 2 } / 2$ 
B)  $t ^ { 2 } / 3$ 
C)  $t ^ { 3 } / 3$ 
D)  $t / 3$ 
E)  $3 t$ E

Question 2

When the Laplace transform is applied to the problem $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { 3 t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 2$ the resulting transformed equation is

Accepted Answer

A)  $\left( s ^ { 2 } + 2 s + 1 \right) Y = - s - 4 + 1 / ( s - 3 )$ 
B)  $\left( s ^ { 2 } + 2 s + 1 \right) Y = s - 4 + 1 / ( s - 3 )$ 
C)  $\left( s ^ { 2 } + 2 s + 1 \right) Y = s + 4 + 1 / ( s + 3 )$ 
D)  $\left( s ^ { 2 } + 2 s + 1 \right) Y = - s - 4 + 1 / ( s + 3 )$ 
E)  $\left( s ^ { 2 } + 2 s + 1 \right) Y = s + 4 + 1 / ( s - 3 )$ 
A)  $\left( s ^ { 2 } + 2 s + 1 \right) Y = - s - 4 + 1 / ( s - 3 )$ 
B)  $\left( s ^ { 2 } + 2 s + 1 \right) Y = s - 4 + 1 / ( s - 3 )$ 
C)  $\left( s ^ { 2 } + 2 s + 1 \right) Y = s + 4 + 1 / ( s + 3 )$ 
D)  $\left( s ^ { 2 } + 2 s + 1 \right) Y = - s - 4 + 1 / ( s + 3 )$ 
E)  $\left( s ^ { 2 } + 2 s + 1 \right) Y = s + 4 + 1 / ( s - 3 )$ E

Question 3

The solution of $f ( t ) = \cos t + \int _ { 0 } ^ { t } e ^ { - \tau } f ( t - \tau ) d \tau$ is

Accepted Answer

A)  $f ( t ) = - \cos t + \sin t$ 
B)  $f ( t ) = \cos t - \sin t$ 
C)  $f ( t ) = \cos t + \sin t$ 
D)  $f ( t ) = - \cos t - \sin t$ 
E)  $f ( t ) = \cos t - 2 \sin t$ 
A)  $f ( t ) = - \cos t + \sin t$ 
B)  $f ( t ) = \cos t - \sin t$ 
C)  $f ( t ) = \cos t + \sin t$ 
D)  $f ( t ) = - \cos t - \sin t$ 
E)  $f ( t ) = \cos t - 2 \sin t$ C

Question 4

The Laplace transform of $e ^ { t } t ^ { 3 } + \sin ( 3 t )$ is

Accepted Answer

A)  $3 / ( s - 1 ) ^ { 3 } + 1 / \left( s ^ { 2 } + 9 \right)$ 
B)  $3 / ( s - 1 ) ^ { 4 } + 1 / \left( s ^ { 2 } + 9 \right)$ 
C)  $6 / ( s - 1 ) ^ { 3 } + 3 / \left( s ^ { 2 } + 9 \right)$ 
D)  $6 / ( s - 1 ) ^ { 4 } + 3 / \left( s ^ { 2 } + 9 \right)$ 
E)  $6 / ( s - 1 ) ^ { 4 } + 9 / \left( s ^ { 2 } + 9 \right)$ 
A)  $3 / ( s - 1 ) ^ { 3 } + 1 / \left( s ^ { 2 } + 9 \right)$ 
B)  $3 / ( s - 1 ) ^ { 4 } + 1 / \left( s ^ { 2 } + 9 \right)$ 
C)  $6 / ( s - 1 ) ^ { 3 } + 3 / \left( s ^ { 2 } + 9 \right)$ 
D)  $6 / ( s - 1 ) ^ { 4 } + 3 / \left( s ^ { 2 } + 9 \right)$ 
E)  $6 / ( s - 1 ) ^ { 4 } + 9 / \left( s ^ { 2 } + 9 \right)$

Question 5

The inverse Laplace transform of $F ( s ) = e ^ { - s } / ( s ( s + 1 ) )$ is

Accepted Answer

A)  $\left( 1 - e ^ { t } \right) u ( t - 1 )$ 
B)  $\left( 1 + e ^ { 1 - t } \right) u ( t - 1 )$ 
C)  $\left( 1 - e ^ { 1 - t } \right) u ( t - 1 )$ 
D)  $\left( 1 - e ^ { t - 1 } \right) u ( t - 1 )$ 
E)  $\left( 1 + e ^ { t - 1 } \right) u ( t - 1 )$ 
A)  $\left( 1 - e ^ { t } \right) u ( t - 1 )$ 
B)  $\left( 1 + e ^ { 1 - t } \right) u ( t - 1 )$ 
C)  $\left( 1 - e ^ { 1 - t } \right) u ( t - 1 )$ 
D)  $\left( 1 - e ^ { t - 1 } \right) u ( t - 1 )$ 
E)  $\left( 1 + e ^ { t - 1 } \right) u ( t - 1 )$

Question 6

The Laplace transform of $t e ^ { - t }$ is

Accepted Answer

A)  $1 / ( s - 1 )$ 
B)  $1 / ( s - 1 ) ^ { 2 }$ 
C)  $1 / ( s + 1 )$ 
D)  $1 / ( s + 1 ) ^ { 2 }$ 
E)  $1 / \left( s ^ { 2 } - 1 \right)$ 
A)  $1 / ( s - 1 )$ 
B)  $1 / ( s - 1 ) ^ { 2 }$ 
C)  $1 / ( s + 1 )$ 
D)  $1 / ( s + 1 ) ^ { 2 }$ 
E)  $1 / \left( s ^ { 2 } - 1 \right)$

Question 7

The Laplace transform of $e ^ { 3 t }$ is

Accepted Answer

A)  $1 / ( s - 3 )$ 
B)  $1 / ( s + 3 )$ 
C)  $1 / ( s - 3 ) ^ { 2 }$ 
D)  $1 / ( s + 3 ) ^ { 2 }$ 
E)  $1 / \left( s ^ { 2 } - 9 \right)$ 
A)  $1 / ( s - 3 )$ 
B)  $1 / ( s + 3 )$ 
C)  $1 / ( s - 3 ) ^ { 2 }$ 
D)  $1 / ( s + 3 ) ^ { 2 }$ 
E)  $1 / \left( s ^ { 2 } - 9 \right)$

Question 8

The Laplace transform of $\int _ { 0 } ^ { t } e ^ { \tau } \sin ( t - \tau ) d \tau$ is

Accepted Answer

A)  $1 / \left( ( s - 1 ) \left( s ^ { 2 } + 1 \right) \right)$ 
B)  $1 / \left( ( s + 1 ) \left( s ^ { 2 } + 1 \right) \right)$ 
C)  $s / \left( ( s - 1 ) \left( s ^ { 2 } + 1 \right) \right)$ 
D)  $s / \left( ( s + 1 ) \left( s ^ { 2 } + 1 \right) \right)$ 
E)  $1 / \left( ( s - 1 ) \left( s ^ { 2 } - 1 \right) \right)$ 
A)  $1 / \left( ( s - 1 ) \left( s ^ { 2 } + 1 \right) \right)$ 
B)  $1 / \left( ( s + 1 ) \left( s ^ { 2 } + 1 \right) \right)$ 
C)  $s / \left( ( s - 1 ) \left( s ^ { 2 } + 1 \right) \right)$ 
D)  $s / \left( ( s + 1 ) \left( s ^ { 2 } + 1 \right) \right)$ 
E)  $1 / \left( ( s - 1 ) \left( s ^ { 2 } - 1 \right) \right)$

Question 9

Let $f ( t ) = \left\{ \begin{array} { l l } 
3 & \text { if } 0 \leq t \leq 2 \
5 - t & \text { if } t > 2
\end{array} \right\}$ . Then $\mathcal { L } \{ f ( t ) \}$ is

Accepted Answer

A)  $3 / s ^ { 2 } - e ^ { - 2 s } / s ^ { 2 }$ 
B)  $3 / s - e ^ { - 2 s } / s ^ { 2 }$ 
C)  $3 / s ^ { 2 } + e ^ { - 2 s } / s ^ { 2 }$ 
D)  $3 / s + e ^ { - 2 s } / s$ 
E)  $3 / s ^ { 2 } + e ^ { 2 s } / s ^ { 2 }$ 
A)  $3 / s ^ { 2 } - e ^ { - 2 s } / s ^ { 2 }$ 
B)  $3 / s - e ^ { - 2 s } / s ^ { 2 }$ 
C)  $3 / s ^ { 2 } + e ^ { - 2 s } / s ^ { 2 }$ 
D)  $3 / s + e ^ { - 2 s } / s$ 
E)  $3 / s ^ { 2 } + e ^ { 2 s } / s ^ { 2 }$

Question 10

The Laplace transform of $t ^ { 3 } e ^ { - t }$ is

Accepted Answer

A)  $1 / ( s + 1 )$ 
B)  $- 1 / ( s + 1 ) ^ { 2 }$ 
C)  $2 / ( s + 1 ) ^ { 3 }$ 
D)  $- 6 / ( s + 1 ) ^ { 4 }$ 
E)  $6 / ( s + 1 ) ^ { 4 }$ 
A)  $1 / ( s + 1 )$ 
B)  $- 1 / ( s + 1 ) ^ { 2 }$ 
C)  $2 / ( s + 1 ) ^ { 3 }$ 
D)  $- 6 / ( s + 1 ) ^ { 4 }$ 
E)  $6 / ( s + 1 ) ^ { 4 }$

Question 11

The solution of the initial value problem in the previous problem is

Accepted Answer

A)  $y = \left( 15 e ^ { t } + 44 t e ^ { t } + e ^ { 3 t } \right) / 16$ 
B)  $y = \left( 15 e ^ { t } - 44 t e ^ { t } + e ^ { 3 t } \right) / 16$ 
C)  $y = \left( 15 e ^ { - t } + 44 t e ^ { - t } + e ^ { 3 t } \right) / 16$ 
D)  $y = \left( 15 e ^ { - t } - 44 t e ^ { - t } + e ^ { 3 t } \right) / 8$ 
E)  $y = \left( 15 e ^ { - t } + 44 t e ^ { - t } + e ^ { 3 t } \right) / 8$ 
A)  $y = \left( 15 e ^ { t } + 44 t e ^ { t } + e ^ { 3 t } \right) / 16$ 
B)  $y = \left( 15 e ^ { t } - 44 t e ^ { t } + e ^ { 3 t } \right) / 16$ 
C)  $y = \left( 15 e ^ { - t } + 44 t e ^ { - t } + e ^ { 3 t } \right) / 16$ 
D)  $y = \left( 15 e ^ { - t } - 44 t e ^ { - t } + e ^ { 3 t } \right) / 8$ 
E)  $y = \left( 15 e ^ { - t } + 44 t e ^ { - t } + e ^ { 3 t } \right) / 8$

Question 12

The solution of the system in the previous problem is

Accepted Answer

A)  $x = 2 e ^ { - 3 t } - e ^ { 2 t } , y = 2 e ^ { - 3 t } - 2 e ^ { 2 t }$ 
B)  $x = 2 e ^ { 3 t } - e ^ { - 2 t } , y = 2 e ^ { 3 t } - 2 e ^ { - 2 t }$ 
C)  $x = 2 e ^ { 3 t } - e ^ { 2 t } , y = 2 e ^ { 3 t } - 2 e ^ { 2 t }$ 
D)  $x = 2 e ^ { - 3 t } - e ^ { - 2 t } , y = 2 e ^ { - 3 t } - 2 e ^ { - 2 t }$ 
E)  $x = 2 e ^ { 3 t } + e ^ { 2 t } , y = 2 e ^ { 3 t } + 2 e ^ { 2 t }$ 
A)  $x = 2 e ^ { - 3 t } - e ^ { 2 t } , y = 2 e ^ { - 3 t } - 2 e ^ { 2 t }$ 
B)  $x = 2 e ^ { 3 t } - e ^ { - 2 t } , y = 2 e ^ { 3 t } - 2 e ^ { - 2 t }$ 
C)  $x = 2 e ^ { 3 t } - e ^ { 2 t } , y = 2 e ^ { 3 t } - 2 e ^ { 2 t }$ 
D)  $x = 2 e ^ { - 3 t } - e ^ { - 2 t } , y = 2 e ^ { - 3 t } - 2 e ^ { - 2 t }$ 
E)  $x = 2 e ^ { 3 t } + e ^ { 2 t } , y = 2 e ^ { 3 t } + 2 e ^ { 2 t }$

Question 13

The inverse Laplace transform of $F ( s ) = e ^ { - 2 s } / s ^ { 2 }$ is

Accepted Answer

A)  $( t - 2 ) \boldsymbol { u } ( t - 2 )$ 
B)  $t u ( t - 2 )$ 
C)  $( t - 2 ) \boldsymbol { u } ( t )$ 
D)  $\left( t ^ { 2 } \right) u ( t - 2 )$ 
E)  $( t - 2 ) ^ { 2 } u ( t - 2 )$ 
A)  $( t - 2 ) \boldsymbol { u } ( t - 2 )$ 
B)  $t u ( t - 2 )$ 
C)  $( t - 2 ) \boldsymbol { u } ( t )$ 
D)  $\left( t ^ { 2 } \right) u ( t - 2 )$ 
E)  $( t - 2 ) ^ { 2 } u ( t - 2 )$

Question 14

The inverse Laplace transform of $F ( s ) = 3 / \left( s ^ { 2 } + 1 \right)$ is

Accepted Answer

A)  $\cos ( 3 t )$ 
B)  $\sin ( 3 t )$ 
C)  $3 \sin ( 3 t )$ 
D)  $3 \sin t$ 
E)  $3 \cos t$ 
A)  $\cos ( 3 t )$ 
B)  $\sin ( 3 t )$ 
C)  $3 \sin ( 3 t )$ 
D)  $3 \sin t$ 
E)  $3 \cos t$

Question 15

The solution of $y ^ { \prime } + 6 y + 9 \int _ { 0 } ^ { t } y ( \tau ) d \tau = 1 , y ( 0 ) = 1$ is

Accepted Answer

A)  $e ^ { 3 t } - 2 t e ^ { 3 t }$ 
B)  $e ^ { 3 t } + 2 t e ^ { 3 t }$ 
C)  $- e ^ { - 3 t } - 2 t e ^ { - 3 t }$ 
D)  $e ^ { - 3 t } - 2 t e ^ { - 3 t }$ 
E)  $e ^ { - 3 t } + 2 t e ^ { - 3 t }$ 
A)  $e ^ { 3 t } - 2 t e ^ { 3 t }$ 
B)  $e ^ { 3 t } + 2 t e ^ { 3 t }$ 
C)  $- e ^ { - 3 t } - 2 t e ^ { - 3 t }$ 
D)  $e ^ { - 3 t } - 2 t e ^ { - 3 t }$ 
E)  $e ^ { - 3 t } + 2 t e ^ { - 3 t }$

Question 16

The inverse Laplace transform of $F ( s ) = ( 2 s - 3 ) / \left( s ^ { 2 } + 1 \right)$ is

Accepted Answer

A)  $2 \cos t + 3 \sin t$ 
B)  $- 2 \cos t - 3 \sin t$ 
C)  $2 \cos t - 3 \sin t$ 
D)  $\cos ( 2 t ) - \sin ( 3 t )$ 
E)  $- \cos ( 2 t ) + \sin ( 3 t )$ 
A)  $2 \cos t + 3 \sin t$ 
B)  $- 2 \cos t - 3 \sin t$ 
C)  $2 \cos t - 3 \sin t$ 
D)  $\cos ( 2 t ) - \sin ( 3 t )$ 
E)  $- \cos ( 2 t ) + \sin ( 3 t )$

Question 17

The inverse Laplace transform of $F ( s ) = 4 / s ^ { 3 }$ is

Accepted Answer

A)  $t ^ { 3 } / 4$ 
B)  $t ^ { 2 } / 4$ 
C)  $t ^ { 2 } / 2$ 
D)  $2 t ^ { 2 }$ 
E)  $3 t ^ { 3 } / 2$ 
A)  $t ^ { 3 } / 4$ 
B)  $t ^ { 2 } / 4$ 
C)  $t ^ { 2 } / 2$ 
D)  $2 t ^ { 2 }$ 
E)  $3 t ^ { 3 } / 2$

Question 18

The inverse Laplace transform of $F ( s ) = \left( 3 s ^ { 2 } + 1 \right) / \left( s ^ { 2 } \cdot \left( s ^ { 2 } + 1 \right) \right)$ is

Accepted Answer

A)  $2 \cos t + t ^ { 2 }$ 
B)  $2 \sin t + t ^ { 2 }$ 
C)  $2 \cos t + t$ 
D)  $2 \sin t + t$ 
E)  $\sin t + 2 t$ 
A)  $2 \cos t + t ^ { 2 }$ 
B)  $2 \sin t + t ^ { 2 }$ 
C)  $2 \cos t + t$ 
D)  $2 \sin t + t$ 
E)  $\sin t + 2 t$

Question 19

A uniform beam of length L has a concentrated load, $w _ { 0 }$ , at $x = L / 2$ . It is embedded at the left end and simply supported at the right end. If $y ( x )$ is the vertical deflection, then the correct differential equation for y, is

Accepted Answer

A)  $E L y ^ { \prime \prime \prime  \prime } = w _ { 0 } \delta ( x - L / 2 )$ 
B)  $y ^ { \prime \prime \prime \prime } = E l w _ { 0 } \delta ( x - L / 2 )$ 
C)  $E L y ^ { \prime \prime } = w _ { 0 } \delta ( x - L / 2 )$ 
D)  $y ^ { \prime \prime } = E l w _ { 0 } \delta ( x - L / 2 )$ 
E)  $E L y ^ { \prime \prime \prime \prime  } = w _ { 0 } \delta ( x + L / 2 )$ 
A)  $E L y ^ { \prime \prime \prime  \prime } = w _ { 0 } \delta ( x - L / 2 )$ 
B)  $y ^ { \prime \prime \prime \prime } = E l w _ { 0 } \delta ( x - L / 2 )$ 
C)  $E L y ^ { \prime \prime } = w _ { 0 } \delta ( x - L / 2 )$ 
D)  $y ^ { \prime \prime } = E l w _ { 0 } \delta ( x - L / 2 )$ 
E)  $E L y ^ { \prime \prime \prime \prime  } = w _ { 0 } \delta ( x + L / 2 )$

Question 20

The Laplace transform of $t ^ { 2 } \cos ( 4 t )$ is

Accepted Answer

A)  $\left( 2 / s ^ { 3 } \right) s / \left( s ^ { 2 } + 16 \right)$ 
B)  $\left( 2 / s ^ { 2 } \right) s / \left( s ^ { 2 } + 16 \right)$ 
C)  $8 / \left( s ^ { 3 } \right) \left( s ^ { 2 } + 16 \right)$ 
D)  $\left( 2 s ^ { 3 } - 32 s \right) / \left( s ^ { 2 } + 16 \right) ^ { 3 }$ 
E)  $\left( 2 s ^ { 3 } - 96 s \right) / \left( s ^ { 2 } + 16 \right) ^ { 3 }$ 
A)  $\left( 2 / s ^ { 3 } \right) s / \left( s ^ { 2 } + 16 \right)$ 
B)  $\left( 2 / s ^ { 2 } \right) s / \left( s ^ { 2 } + 16 \right)$ 
C)  $8 / \left( s ^ { 3 } \right) \left( s ^ { 2 } + 16 \right)$ 
D)  $\left( 2 s ^ { 3 } - 32 s \right) / \left( s ^ { 2 } + 16 \right) ^ { 3 }$ 
E)  $\left( 2 s ^ { 3 } - 96 s \right) / \left( s ^ { 2 } + 16 \right) ^ { 3 }$

The inverse Laplace transform of $F ( s ) = 3 / s ^ { 2 }$ is

When the Laplace transform is applied to the problem $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { 3 t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 2$ the resulting transformed equation is

The solution of $f ( t ) = \cos t + \int _ { 0 } ^ { t } e ^ { - \tau } f ( t - \tau ) d \tau$ is

The Laplace transform of $e ^ { t } t ^ { 3 } + \sin ( 3 t )$ is

The inverse Laplace transform of $F ( s ) = e ^ { - s } / ( s ( s + 1 ) )$ is

The Laplace transform of $t e ^ { - t }$ is

The Laplace transform of $e ^ { 3 t }$ is

The Laplace transform of $\int _ { 0 } ^ { t } e ^ { \tau } \sin ( t - \tau ) d \tau$ is

Let $f ( t ) = \left\{ \begin{array} { l l } 3 & \text { if } 0 \leq t \leq 2 \\5 - t & \text { if } t > 2\end{array} \right\}$ . Then $\mathcal { L } \{ f ( t ) \}$ is

The Laplace transform of $t ^ { 3 } e ^ { - t }$ is

The solution of the initial value problem in the previous problem is

The solution of the system in the previous problem is

The inverse Laplace transform of $F ( s ) = e ^ { - 2 s } / s ^ { 2 }$ is

The inverse Laplace transform of $F ( s ) = 3 / \left( s ^ { 2 } + 1 \right)$ is

The solution of $y ^ { \prime } + 6 y + 9 \int _ { 0 } ^ { t } y ( \tau ) d \tau = 1 , y ( 0 ) = 1$ is

The inverse Laplace transform of $F ( s ) = ( 2 s - 3 ) / \left( s ^ { 2 } + 1 \right)$ is

The inverse Laplace transform of $F ( s ) = 4 / s ^ { 3 }$ is

The inverse Laplace transform of $F ( s ) = \left( 3 s ^ { 2 } + 1 \right) / \left( s ^ { 2 } \cdot \left( s ^ { 2 } + 1 \right) \right)$ is

A uniform beam of length L has a concentrated load, $w _ { 0 }$ , at $x = L / 2$ . It is embedded at the left end and simply supported at the right end. If $y ( x )$ is the vertical deflection, then the correct differential equation for y, is

The Laplace transform of $t ^ { 2 } \cos ( 4 t )$ is

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

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Exam 7: Laplace Transform

The inverse Laplace transform of F(s)=3/s2F ( s ) = 3 / s ^ { 2 }F(s)=3/s2 is

When the Laplace transform is applied to the problem y′′+2y′+y=e3t,y(0)=1,y′(0)=2y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { 3 t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 2y′′+2y′+y=e3t,y(0)=1,y′(0)=2 the resulting transformed equation is

The solution of f(t)=cos⁡t+∫0te−τf(t−τ)dτf ( t ) = \cos t + \int _ { 0 } ^ { t } e ^ { - \tau } f ( t - \tau ) d \tauf(t)=cost+∫0t​e−τf(t−τ)dτ is

The Laplace transform of ett3+sin⁡(3t)e ^ { t } t ^ { 3 } + \sin ( 3 t )ett3+sin(3t) is

The inverse Laplace transform of F(s)=e−s/(s(s+1))F ( s ) = e ^ { - s } / ( s ( s + 1 ) )F(s)=e−s/(s(s+1)) is

The Laplace transform of te−tt e ^ { - t }te−t is

The Laplace transform of e3te ^ { 3 t }e3t is

The Laplace transform of ∫0teτsin⁡(t−τ)dτ\int _ { 0 } ^ { t } e ^ { \tau } \sin ( t - \tau ) d \tau∫0t​eτsin(t−τ)dτ is

Let f(t)={3 if 0≤t≤25−t if t>2}f ( t ) = \left\{ \begin{array} { l l } 3 & \text { if } 0 \leq t \leq 2 \\5 - t & \text { if } t > 2\end{array} \right\}f(t)={35−t​ if 0≤t≤2 if t>2​} . Then L{f(t)}\mathcal { L } \{ f ( t ) \}L{f(t)} is

The Laplace transform of t3e−tt ^ { 3 } e ^ { - t }t3e−t is

The solution of the initial value problem in the previous problem is

The solution of the system in the previous problem is

The inverse Laplace transform of F(s)=e−2s/s2F ( s ) = e ^ { - 2 s } / s ^ { 2 }F(s)=e−2s/s2 is

The inverse Laplace transform of F(s)=3/(s2+1)F ( s ) = 3 / \left( s ^ { 2 } + 1 \right)F(s)=3/(s2+1) is

The solution of y′+6y+9∫0ty(τ)dτ=1,y(0)=1y ^ { \prime } + 6 y + 9 \int _ { 0 } ^ { t } y ( \tau ) d \tau = 1 , y ( 0 ) = 1y′+6y+9∫0t​y(τ)dτ=1,y(0)=1 is

The inverse Laplace transform of F(s)=(2s−3)/(s2+1)F ( s ) = ( 2 s - 3 ) / \left( s ^ { 2 } + 1 \right)F(s)=(2s−3)/(s2+1) is

The inverse Laplace transform of F(s)=4/s3F ( s ) = 4 / s ^ { 3 }F(s)=4/s3 is

The inverse Laplace transform of F(s)=(3s2+1)/(s2⋅(s2+1))F ( s ) = \left( 3 s ^ { 2 } + 1 \right) / \left( s ^ { 2 } \cdot \left( s ^ { 2 } + 1 \right) \right)F(s)=(3s2+1)/(s2⋅(s2+1)) is

A uniform beam of length L has a concentrated load, w0w _ { 0 }w0​ , at x=L/2x = L / 2x=L/2 . It is embedded at the left end and simply supported at the right end. If y(x)y ( x )y(x) is the vertical deflection, then the correct differential equation for y, is

The Laplace transform of t2cos⁡(4t)t ^ { 2 } \cos ( 4 t )t2cos(4t) is

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

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The inverse Laplace transform of $F ( s ) = 3 / s ^ { 2 }$ is

When the Laplace transform is applied to the problem $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { 3 t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 2$ the resulting transformed equation is

The solution of $f ( t ) = \cos t + \int _ { 0 } ^ { t } e ^ { - \tau } f ( t - \tau ) d \tau$ is

The Laplace transform of $e ^ { t } t ^ { 3 } + \sin ( 3 t )$ is

The inverse Laplace transform of $F ( s ) = e ^ { - s } / ( s ( s + 1 ) )$ is

The Laplace transform of $t e ^ { - t }$ is

The Laplace transform of $e ^ { 3 t }$ is

The Laplace transform of $\int _ { 0 } ^ { t } e ^ { \tau } \sin ( t - \tau ) d \tau$ is

Let $f ( t ) = \left\{ \begin{array} { l l } 3 & \text { if } 0 \leq t \leq 2 \\5 - t & \text { if } t > 2\end{array} \right\}$ . Then $\mathcal { L } \{ f ( t ) \}$ is

The Laplace transform of $t ^ { 3 } e ^ { - t }$ is

The inverse Laplace transform of $F ( s ) = e ^ { - 2 s } / s ^ { 2 }$ is

The inverse Laplace transform of $F ( s ) = 3 / \left( s ^ { 2 } + 1 \right)$ is

The solution of $y ^ { \prime } + 6 y + 9 \int _ { 0 } ^ { t } y ( \tau ) d \tau = 1 , y ( 0 ) = 1$ is

The inverse Laplace transform of $F ( s ) = ( 2 s - 3 ) / \left( s ^ { 2 } + 1 \right)$ is

The inverse Laplace transform of $F ( s ) = 4 / s ^ { 3 }$ is

The inverse Laplace transform of $F ( s ) = \left( 3 s ^ { 2 } + 1 \right) / \left( s ^ { 2 } \cdot \left( s ^ { 2 } + 1 \right) \right)$ is

A uniform beam of length L has a concentrated load, $w _ { 0 }$ , at $x = L / 2$ . It is embedded at the left end and simply supported at the right end. If $y ( x )$ is the vertical deflection, then the correct differential equation for y, is

The Laplace transform of $t ^ { 2 } \cos ( 4 t )$ is