Question 1

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

Accepted Answer

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

Question 2

When the Laplace transform is applied to the system $\begin{array} { l } 
\frac { d x } { d t } = 3 x - y \
\frac { d y } { d t } = x + y \
x ( 0 ) = 2 , y ( 0 ) = 1
\end{array}$ the resulting transformed system is

Accepted Answer

A)  $s X = 3 X - Y - 2 , s Y - 1 = X + Y$ 
B)  $s X - 2 = 3 X - Y , s Y = X + Y - 1$ 
C)  $s X = 3 X - Y - 2 , s Y = X + Y - 1$ 
D)  $s X - 2 = 3 X - Y , s Y - 1 = X + Y$ 
E)  $s X + 2 = 3 X - Y , s Y + 1 = X + Y$ 
A)  $s X = 3 X - Y - 2 , s Y - 1 = X + Y$ 
B)  $s X - 2 = 3 X - Y , s Y = X + Y - 1$ 
C)  $s X = 3 X - Y - 2 , s Y = X + Y - 1$ 
D)  $s X - 2 = 3 X - Y , s Y - 1 = X + Y$ 
E)  $s X + 2 = 3 X - Y , s Y + 1 = X + Y$

Question 3

The Laplace transform of $\cos t$ is

Accepted Answer

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

Question 4

The solution of the system in the previous problem is

Accepted Answer

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

Question 5

The Laplace transform of $\int _ { 0 } ^ { t } \tau ^ { 2 } \cos ( t - \tau ) d \tau$ is

Accepted Answer

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

Question 6

The inverse Laplace transform of $F ( s ) = ( 5 s + 2 ) / \left( s ^ { 2 } \cdot \left( s ^ { 2 } - 1 \right) \right)$ is

Accepted Answer

A)  $- 5 - 2 t + 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$ 
B)  $5 - 2 t - 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$ 
C)  $5 - 2 t + 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$ 
D)  $- 5 + 2 t - 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$ 
E)  $- 5 + 2 t + 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$ 
A)  $- 5 - 2 t + 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$ 
B)  $5 - 2 t - 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$ 
C)  $5 - 2 t + 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$ 
D)  $- 5 + 2 t - 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$ 
E)  $- 5 + 2 t + 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$

Question 7

When the Laplace transform is applied to the system $\begin{array} { l } 
\frac { d x } { d t } = 4 x - y \
\frac { d y } { d t } = 2 x + y \
x ( 0 ) = 1 , y ( 0 ) = 0
\end{array}$ the resulting transformed system is

Accepted Answer

A)  $s X = 4 X - Y , s Y - 1 = 2 X + Y$ 
B)  $s X = 4 X - Y - 1 , s Y = 2 X + Y$ 
C)  $s X = 4 X - Y , s Y = 2 X + Y - 1$ 
D)  $s X = 4 X - Y , s Y = 2 X + Y$ 
E)  $s X - 1 = 4 X - Y , s Y = 2 X + Y$ 
A)  $s X = 4 X - Y , s Y - 1 = 2 X + Y$ 
B)  $s X = 4 X - Y - 1 , s Y = 2 X + Y$ 
C)  $s X = 4 X - Y , s Y = 2 X + Y - 1$ 
D)  $s X = 4 X - Y , s Y = 2 X + Y$ 
E)  $s X - 1 = 4 X - Y , s Y = 2 X + Y$

Question 8

The Laplace transform of $\sin ( 2 t )$ is

Accepted Answer

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

Question 9

The Laplace transform of $te ^ { t }$ 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 10

A uniform beam of length L has a concentrated load, $w _ { 0 }$ , at $x = L / 2$ . It is embedded at the left end and free at the right end. The correct initial value problem for the vertical deflection, $y ( x )$ , at a distance x from the embedded end is

Accepted Answer

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

Question 11

The Laplace transform of $t ^ { 2 } \sin ( 3 t )$ is

Accepted Answer

A)  $\left( 18 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$ 
B)  $\left( 18 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 2 }$ 
C)  $\left( 30 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 2 }$ 
D)  $\left( 30 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$ 
E)  $\left( 12 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$ 
A)  $\left( 18 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$ 
B)  $\left( 18 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 2 }$ 
C)  $\left( 30 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 2 }$ 
D)  $\left( 30 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$ 
E)  $\left( 12 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$

Question 12

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

Accepted Answer

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

Let $f ( t ) = \left\{ \begin{array} { l l } 1 & \text { if } 0 \leq t \leq 3 \\t - 2 & \text { if } t > 3\end{array} \right\}$ . Then $\mathcal { L } \{ f ( t ) \}$ is

When the Laplace transform is applied to the system =3x-y =x+y x(0)=2,y(0)=1 the resulting transformed system is

The Laplace transform of $\cos t$ is

The solution of the system in the previous problem is

The Laplace transform of $\int _ { 0 } ^ { t } \tau ^ { 2 } \cos ( t - \tau ) d \tau$ is

The inverse Laplace transform of $F ( s ) = ( 5 s + 2 ) / \left( s ^ { 2 } \cdot \left( s ^ { 2 } - 1 \right) \right)$ is

When the Laplace transform is applied to the system =4x-y =2x+y x(0)=1,y(0)=0 the resulting transformed system is

The Laplace transform of $\sin ( 2 t )$ is

The Laplace transform of $te ^ { t }$ 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 free at the right end. The correct initial value problem for the vertical deflection, $y ( x )$ , at a distance x from the embedded end is

The Laplace transform of $t ^ { 2 } \sin ( 3 t )$ is

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

Mathematical Problems and Solutions

Filters

Exam 7: Laplace Transform

Let f(t)={1 if 0≤t≤3t−2 if t>3}f ( t ) = \left\{ \begin{array} { l l } 1 & \text { if } 0 \leq t \leq 3 \\t - 2 & \text { if } t > 3\end{array} \right\}f(t)={1t−2​ if 0≤t≤3 if t>3​} . Then L{f(t)}\mathcal { L } \{ f ( t ) \}L{f(t)} is

When the Laplace transform is applied to the system =3x-y =x+y x(0)=2,y(0)=1 the resulting transformed system is

The Laplace transform of cos⁡t\cos tcost is

The solution of the system in the previous problem is

The Laplace transform of ∫0tτ2cos⁡(t−τ)dτ\int _ { 0 } ^ { t } \tau ^ { 2 } \cos ( t - \tau ) d \tau∫0t​τ2cos(t−τ)dτ is

The inverse Laplace transform of F(s)=(5s+2)/(s2⋅(s2−1))F ( s ) = ( 5 s + 2 ) / \left( s ^ { 2 } \cdot \left( s ^ { 2 } - 1 \right) \right)F(s)=(5s+2)/(s2⋅(s2−1)) is

When the Laplace transform is applied to the system =4x-y =2x+y x(0)=1,y(0)=0 the resulting transformed system is

The Laplace transform of sin⁡(2t)\sin ( 2 t )sin(2t) is

The Laplace transform of tette ^ { t }tet 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 free at the right end. The correct initial value problem for the vertical deflection, y(x)y ( x )y(x) , at a distance x from the embedded end is

The Laplace transform of t2sin⁡(3t)t ^ { 2 } \sin ( 3 t )t2sin(3t) is

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

Mathematical Problems and Solutions

Filters

Let $f ( t ) = \left\{ \begin{array} { l l } 1 & \text { if } 0 \leq t \leq 3 \\t - 2 & \text { if } t > 3\end{array} \right\}$ . Then $\mathcal { L } \{ f ( t ) \}$ is

The Laplace transform of $\cos t$ is

The Laplace transform of $\int _ { 0 } ^ { t } \tau ^ { 2 } \cos ( t - \tau ) d \tau$ is

The inverse Laplace transform of $F ( s ) = ( 5 s + 2 ) / \left( s ^ { 2 } \cdot \left( s ^ { 2 } - 1 \right) \right)$ is

The Laplace transform of $\sin ( 2 t )$ is

The Laplace transform of $te ^ { t }$ 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 free at the right end. The correct initial value problem for the vertical deflection, $y ( x )$ , at a distance x from the embedded end is

The Laplace transform of $t ^ { 2 } \sin ( 3 t )$ is

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