Question 1

The inverse Laplace transform of $F ( s ) = \left( 3 s ^ { 2 } + 1 \right) / \left( s ^ { 2 } \cdot \left( s ^ { 2 } + 1 \right) \right)$ is&#10;A) $2 \cos t + t ^ { 2 }$&#10;B) $2 \sin t + t ^ { 2 }$&#10;C) $2 \cos t + t$&#10;D) $2 \sin t + t$&#10;E) $\sin t + 2 t$

Accepted Answer

The given function $F(s) = \frac{3s^2 + 1}{s^2(s^2 + 1)}$ can be decomposed into partial fractions to facilitate finding its inverse Laplace transform. The decomposition leads to a term that corresponds to the inverse Laplace transform of a constant (which gives a ramp function, $t$) and a term that corresponds to the inverse Laplace transform of $\frac{1}{s^2 + 1}$, which is $\sin(t)$. Thus, the correct answer involves both $t$ and $\sin(t)$, making option D ($2 \sin t + t$) the correct choice.

Question 2

The inverse Laplace transform of $F ( s ) = e ^ { - s } / ( s ( s + 1 ) )$ is&#10;A) $\left( 1 - e ^ { t } \right) u ( t - 1 )$&#10;B) $\left( 1 + e ^ { 1 - t } \right) u ( t - 1 )$&#10;C) $\left( 1 - e ^ { 1 - t } \right) u ( t - 1 )$&#10;D) $\left( 1 - e ^ { t - 1 } \right) u ( t - 1 )$&#10;E) $\left( 1 + e ^ { t - 1 } \right) u ( t - 1 )$

Accepted Answer

The term $e^{-s}$ in the Laplace domain corresponds to a time shift of 1 in the time domain, and the inverse Laplace transform of $1/(s(s+1))$ is $1-e^{-t}$. Applying the time shift results in $\left(1-e^{1-t}\right)u(t-1)$, where $u(t-1)$ is the unit step function that shifts the response to start at $t=1$.

Question 3

The Laplace transform of $te ^ { t }$ is&#10;A) $1 / ( s - 1 )$&#10;B) $$1 / ( s - 1 ) ^ { 2 }$$&#10;C) $1 / ( s + 1 )$&#10;D) $1 / ( s + 1 ) ^ { 2 }$&#10;E) $1 / \left( s ^ { 2 } - 1 \right)$

Accepted Answer

The Laplace transform of $te^t$ is obtained by applying the formula for the Laplace transform of $t^n e^{at}$, which is $\frac{n!}{(s-a)^{n+1}}$ for $n=0, 1, 2, \ldots$ and $a$ being a constant. Here, $n=1$ and $a=1$, so the transform is $\frac{1!}{(s-1)^{2}}$, which matches option B.

Question 4

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

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

Accepted Answer

The answer of When the Laplace transform is applied to...

Question 5

The Laplace transform of $e ^ { - t }$ is&#10;A) $1 / ( s - 1 )$&#10;B) $1 / ( s + 1 )$&#10;C) $1 / ( s - 1 ) ^ { 2 }$&#10;D) $1 / ( s + 1 ) ^ { 2 }$&#10;E) $1 / \left( s ^ { 2 } - 1 \right)$

Accepted Answer

The Laplace transform of $e^{-t}$ is given by $L\{e^{-t}\} = \frac{1}{s+1}$, where $s$ is a complex number and the real part of $s$ must be greater than -1 for the transform to exist. This is derived from the general formula for the Laplace transform of $e^{at}$, which is $\frac{1}{s-a}$, by setting $a = -1$.

Question 6

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

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 )

Accepted Answer

The correct differential equation for the vertical deflection $y(x)$ of a beam under a concentrated load is given by the Euler-Bernoulli beam equation, which in this case is $E I y^{''''} = w_0 \delta(x - L/2)$, where $E$ is the modulus of elasticity, $I$ is the moment of inertia of the beam's cross-section, and $\delta$ is the Dirac delta function representing the concentrated load. Option A correctly represents this relationship, albeit with a slight notation difference, assuming $L$ in the option might be a typo or a simplification for $EI$.

Question 7

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

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

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Accepted Answer

The answer of When the Laplace transform is applied to...

Question 8

The inverse Laplace transform of $F ( s ) = 4 / s ^ { 3 }$ is&#10;A) $t ^ { 3 } / 4$&#10;B) $t ^ { 2 } / 4$&#10;C) $t ^ { 2 } / 2$&#10;D) $2 t ^ { 2 }$&#10;E) $3 t ^ { 3 } / 2$

Accepted Answer

The inverse Laplace transform of $4/s^3$ is $2t^2$, as the Laplace transform of $t^n$ is $n!/s^{n+1}$. Here, $n=2$, so the inverse is $2t^2$.

Question 9

The solution of the system in the previous problem is&#10;A) $x = 2 e ^ { - 2 t } + t e ^ { - 2 t } , y = e ^ { - 2 t } + t e ^ { - 2 t }$&#10;B) $x = 2 e ^ { - 2 t } + t e ^ { - 2 t } , y = e ^ { - 2 t } - t e ^ { - 2 t }$&#10;C) $$x = 2 e ^ { 2 t } + t e ^ { 2 t } , y = e ^ { 2 t } + t e ^ { 2 t }$$&#10;D) $x = 2 e ^ { 2 t } - t e ^ { 2 t } , y = e ^ { 2 t } - t e ^ { 2 t }$&#10;E) $x = 2 e ^ { 2 t } + t e ^ { 2 t } , y = e ^ { 2 t } - t e ^ { 2 t }$

Accepted Answer

The answer of The solution of the system in the...

Question 10

The Laplace transform of $\sin ( 2 t )$ is&#10;A) $$1 / ( s - 2 )$$&#10;B) $s / \left( s ^ { 2 } - 4 \right)$&#10;C) $s / \left( s ^ { 2 } + 4 \right)$&#10;D) $2 / \left( s ^ { 2 } - 4 \right)$&#10;E) $2 / \left( s ^ { 2 } + 4 \right)$

Accepted Answer

The Laplace transform of $\sin(at)$ is $a / (s^2 + a^2)$. For $\sin(2t)$, $a = 2$, so the transform is $2 / (s^2 + 4)$.

Question 11

Let $f ( t ) = \left\{ \begin{array} { l l } &#10;1 & \text { if } 0 \leq t \leq 3 \&#10;t - 2 & \text { if } t > 3&#10;\end{array} \right\}$ . Then $\mathcal { L } \{ f ( t ) \}$ is&#10;A) $1 / s - e ^ { - 3 s } / s ^ { 2 }$&#10;B) $1 / s + e ^ { 3 s } / s$&#10;C) $1 / s + e ^ { - 3 s } / s$&#10;D) $1 / s + e ^ { 3 s } / s ^ { 2 }$&#10;E) $1 / s + e ^ { - 3 s } / s ^ { 2 }$

Accepted Answer

The answer of Let \(f ( t ) = \left\{...

Question 12

The Laplace transform of $\cos t$ is&#10;A) $1 / ( s - 1 )$&#10;B) $s / \left( s ^ { 2 } - 1 \right)$&#10;C) $s / \left( s ^ { 2 } + 1 \right)$&#10;D) $1 / \left( s ^ { 2 } - 1 \right)$&#10;E) $1 / \left( s ^ { 2 } + 1 \right)$

Accepted Answer

The answer of The Laplace transform of $\cos t$ is&#10;A)...

Question 13

The Laplace transform of $\int _ { 0 } ^ { t } \tau ^ { 2 } \cos ( t - \tau ) d \tau$ is&#10;A) $2 / \left( s ^ { 2 } \left( s ^ { 2 } + 1 \right) \right)$&#10;B) $1 / \left( s ^ { 2 } \left( s ^ { 2 } + 1 \right) \right)$&#10;C) $2 / \left( s ^ { 3 } \left( s ^ { 2 } + 1 \right) \right)$&#10;D) $6 / \left( s ^ { 2 } \left( s ^ { 2 } + 1 \right) \right)$&#10;E) $6 / \left( s ^ { 3 } \left( s ^ { 2 } + 1 \right) \right)$

Accepted Answer

The answer of The Laplace transform of \(\int _ {...

Question 14

The Laplace transform of $t ^ { 3 } e ^ { - t }$ is&#10;A) $1 / ( s + 1 )$&#10;B) $- 1 / ( s + 1 ) ^ { 2 }$&#10;C) $2 / ( s + 1 ) ^ { 3 }$&#10;D) $- 6 / ( s + 1 ) ^ { 4 }$&#10;E) $6 / ( s + 1 ) ^ { 4 }$

Accepted Answer

The answer of The Laplace transform of \(t ^ {...

Question 15

The solution of the initial value problem in the previous problem is&#10;A) $y = \left( 15 e ^ { t } + 44 t e ^ { t } + e ^ { 3 t } \right) / 16$&#10;B) $y = \left( 15 e ^ { t } - 44 t e ^ { t } + e ^ { 3 t } \right) / 16$&#10;C) $y = \left( 15 e ^ { - t } + 44 t e ^ { - t } + e ^ { 3 t } \right) / 16$&#10;D) $y = \left( 15 e ^ { - t } - 44 t e ^ { - t } + e ^ { 3 t } \right) / 8$&#10;E) $y = \left( 15 e ^ { - t } + 44 t e ^ { - t } + e ^ { 3 t } \right) / 8$

Accepted Answer

The answer of The solution of the initial value problem...

Question 16

The solution of $f ( t ) = \cos t + \int _ { 0 } ^ { t } e ^ { - \tau } f ( t - \tau ) d \tau$ is&#10;A) $f ( t ) = - \cos t + \sin t$&#10;B) $f ( t ) = \cos t - \sin t$&#10;C) $f ( t ) = \cos t + \sin t$&#10;D) $f ( t ) = - \cos t - \sin t$&#10;E) $f ( t ) = \cos t - 2 \sin t$

Accepted Answer

The answer of The solution of \(f ( t )...

Question 17

The Laplace transform of $e ^ { 3 t }$ is&#10;A) $1 / ( s - 3 )$&#10;B) $1 / ( s + 3 )$&#10;C) $1 / ( s - 3 ) ^ { 2 }$&#10;D) $1 / ( s + 3 ) ^ { 2 }$&#10;E) $1 / \left( s ^ { 2 } - 9 \right)$

Accepted Answer

The answer of The Laplace transform of \(e ^ {...

Question 18

The Laplace transform of $e ^ { t } t ^ { 3 } + \sin ( 3 t )$ is&#10;A) $3 / ( s - 1 ) ^ { 3 } + 1 / \left( s ^ { 2 } + 9 \right)$&#10;B) $3 / ( s - 1 ) ^ { 4 } + 1 / \left( s ^ { 2 } + 9 \right)$&#10;C) $6 / ( s - 1 ) ^ { 3 } + 3 / \left( s ^ { 2 } + 9 \right)$&#10;D) $6 / ( s - 1 ) ^ { 4 } + 3 / \left( s ^ { 2 } + 9 \right)$&#10;E) $6 / ( s - 1 ) ^ { 4 } + 9 / \left( s ^ { 2 } + 9 \right)$

Accepted Answer

The answer of The Laplace transform of \(e ^ {...

Question 19

The inverse Laplace transform of $F ( s ) = ( 2 s - 3 ) / \left( s ^ { 2 } + 1 \right)$ is&#10;A) $2 \cos t + 3 \sin t$&#10;B) $- 2 \cos t - 3 \sin t$&#10;C) $2 \cos t - 3 \sin t$&#10;D) $\cos ( 2 t ) - \sin ( 3 t )$&#10;E) $- \cos ( 2 t ) + \sin ( 3 t )$

Accepted Answer

The answer of The inverse Laplace transform of \(F (...

Question 20

Let $f ( t ) = \left\{ \begin{array} { l l } &#10;3 & \text { if } 0 \leq t \leq 2 \&#10;5 - t & \text { if } t > 2&#10;\end{array} \right\}$ . Then $\mathcal { L } \{ f ( t ) \}$ is&#10;A) $3 / s ^ { 2 } - e ^ { - 2 s } / s ^ { 2 }$&#10;B) $3 / s - e ^ { - 2 s } / s ^ { 2 }$&#10;C) $3 / s ^ { 2 } + e ^ { - 2 s } / s ^ { 2 }$&#10;D) $3 / s + e ^ { - 2 s } / s$&#10;E) $3 / s ^ { 2 } + e ^ { 2 s } / s ^ { 2 }$

Accepted Answer

The answer of Let \(f ( t ) = \left\{...

Question 21

The Laplace transform of $t e ^ { - t }$ is&#10;A) $1 / ( s - 1 )$&#10;B) $1 / ( s - 1 ) ^ { 2 }$&#10;C) $1 / ( s + 1 )$&#10;D) $1 / ( s + 1 ) ^ { 2 }$&#10;E) $1 / \left( s ^ { 2 } - 1 \right)$

Accepted Answer

The answer of The Laplace transform of \(t e ^...

Question 22

The inverse Laplace transform of $F ( s ) = 3 / s ^ { 2 }$ is&#10;A) $t ^ { 2 } / 2$&#10;B) $t ^ { 2 } / 3$&#10;C) $t ^ { 3 } / 3$&#10;D) $t / 3$&#10;E) $3 t$

Accepted Answer

The answer of The inverse Laplace transform of \(F (...

Question 23

The solution of the system in the previous problem is&#10;A) $x = 2 e ^ { - 3 t } - e ^ { 2 t } , y = 2 e ^ { - 3 t } - 2 e ^ { 2 t }$&#10;B) $x = 2 e ^ { 3 t } - e ^ { - 2 t } , y = 2 e ^ { 3 t } - 2 e ^ { - 2 t }$&#10;C) $x = 2 e ^ { 3 t } - e ^ { 2 t } , y = 2 e ^ { 3 t } - 2 e ^ { 2 t }$&#10;D) $x = 2 e ^ { - 3 t } - e ^ { - 2 t } , y = 2 e ^ { - 3 t } - 2 e ^ { - 2 t }$&#10;E) $x = 2 e ^ { 3 t } + e ^ { 2 t } , y = 2 e ^ { 3 t } + 2 e ^ { 2 t }$

Accepted Answer

The answer of The solution of the system in the...

Question 24

The inverse Laplace transform of $F ( s ) = 3 / \left( s ^ { 2 } + 1 \right)$ is&#10;A) $\cos ( 3 t )$&#10;B) $\sin ( 3 t )$&#10;C) $3 \sin ( 3 t )$&#10;D) $3 \sin t$&#10;E) $3 \cos t$

Accepted Answer

The answer of The inverse Laplace transform of \(F (...

Question 25

The inverse Laplace transform of $F ( s ) = ( 5 s + 2 ) / \left( s ^ { 2 } \cdot \left( s ^ { 2 } - 1 \right) \right)$ is&#10;A) $- 5 - 2 t + 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$&#10;B) $5 - 2 t - 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$&#10;C) $5 - 2 t + 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$&#10;D) $- 5 + 2 t - 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$&#10;E) $- 5 + 2 t + 7 e ^ { t } / 2 + 3 e ^ { - t } / 2$

Accepted Answer

The answer of The inverse Laplace transform of \(F (...

Question 26

The Laplace transform of $t ^ { 2 } \sin ( 3 t )$ is&#10;A) $\left( 18 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$&#10;B) $\left( 18 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 2 }$&#10;C) $\left( 30 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 2 }$&#10;D) $\left( 30 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$&#10;E) $\left( 12 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$

Accepted Answer

The answer of The Laplace transform of \(t ^ {...

Question 27

The Laplace transform of $t ^ { 2 } \cos ( 4 t )$ is&#10;A) $\left( 2 / s ^ { 3 } \right) s / \left( s ^ { 2 } + 16 \right)$&#10;B) $\left( 2 / s ^ { 2 } \right) s / \left( s ^ { 2 } + 16 \right)$&#10;C) $8 / \left( s ^ { 3 } \right) \left( s ^ { 2 } + 16 \right)$&#10;D) $\left( 2 s ^ { 3 } - 32 s \right) / \left( s ^ { 2 } + 16 \right) ^ { 3 }$&#10;E) $\left( 2 s ^ { 3 } - 96 s \right) / \left( s ^ { 2 } + 16 \right) ^ { 3 }$

Accepted Answer

The answer of The Laplace transform of \(t ^ {...

Question 28

The Laplace transform of $\int _ { 0 } ^ { t } e ^ { \tau } \sin ( t - \tau ) d \tau$ is&#10;A) $1 / \left( ( s - 1 ) \left( s ^ { 2 } + 1 \right) \right)$&#10;B) $1 / \left( ( s + 1 ) \left( s ^ { 2 } + 1 \right) \right)$&#10;C) $s / \left( ( s - 1 ) \left( s ^ { 2 } + 1 \right) \right)$&#10;D) $s / \left( ( s + 1 ) \left( s ^ { 2 } + 1 \right) \right)$&#10;E) $1 / \left( ( s - 1 ) \left( s ^ { 2 } - 1 \right) \right)$

Accepted Answer

The answer of The Laplace transform of \(\int _ {...

Question 29

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

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

Accepted Answer

The answer of When the Laplace transform is applied to...

Question 30

The solution of $y ^ { \prime } + 6 y + 9 \int _ { 0 } ^ { t } y ( \tau ) d \tau = 1 , y ( 0 ) = 1$ is&#10;A) $e ^ { 3 t } - 2 t e ^ { 3 t }$&#10;B) $e ^ { 3 t } + 2 t e ^ { 3 t }$&#10;C) $- e ^ { - 3 t } - 2 t e ^ { - 3 t }$&#10;D) $e ^ { - 3 t } - 2 t e ^ { - 3 t }$&#10;E) $e ^ { - 3 t } + 2 t e ^ { - 3 t }$

Accepted Answer

The answer of The solution of \(y ^ { \prime...

Question 31

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

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

Accepted Answer

The answer of A uniform beam of length L has...

Question 32

The inverse Laplace transform of $F ( s ) = e ^ { - 2 s } / s ^ { 2 }$ is&#10;A) $( t - 2 ) \boldsymbol { u } ( t - 2 )$&#10;B) $t u ( t - 2 )$&#10;C) $( t - 2 ) \boldsymbol { u } ( t )$&#10;D) $\left( t ^ { 2 } \right) u ( t - 2 )$&#10;E) $( t - 2 ) ^ { 2 } u ( t - 2 )$

Accepted Answer

The answer of The inverse Laplace transform of \(F (...

Deck 7: Laplace Transform