Laplace transform

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Problem: Find the Laplace transformation of the periodic function \( f(t)=B|\sin \omega t| \).

10.4.1 Laplace transform

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Problem: Find the Laplace transform of the function: \[ f(t)=\left\{\begin{array}{ll} 0, & t<0 \\ -2, & 0 \leq t<2 \\ 0, & 2 \leq t<3 \\ \frac{1}{2}, & 3 \leq t<5 \\ 0, & 5 \geq t \end{array}\right. \]

10.4.2 Laplace transform

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Problem: Find the Laplace transform of the function: \[ F(P)=\frac{2 P}{\left(P^{2}+4 P+8\right)^{2}} . \]

10.4.3 Laplace transform

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Problem: Find the Laplace transform \( F(p) \) according to the given signal: \[ f(t)=\int_{0}^{t} \tau \exp \left(-\frac{t-\tau}{2}\right) d \tau \text {. } \]

10.4.4 Laplace transform

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Problem: Find the Laplace transform \( F(p) \) by the given signal \( f(t)=\theta(t-1) t e^{t} \).

10.4.5 Laplace transform

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Problem: Find the Laplace transform \( F(p) \) by the given signal: \[ f(t)=\theta\left(t-\frac{\pi}{4}\right) \sin t \text {. } \]

10.4.6 Laplace transform

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Problem: Find the signal \( f(t) \) by the given Laplace transform: \[ F(p)=\frac{1}{p-2}+\frac{e^{-p}}{p}+\frac{3 e^{-4 p}}{p^{2}+9} \]

10.4.7 Laplace transform

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Problem: Find the Laplace transform of the signal by the given graph:

10.4.8 Laplace transform

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Problem: Find the signal of the Laplace transform: \[ \frac{p-8}{(p+1)\left(p^{2}+2 p+10\right)} \text {. } \]

10.4.9 Laplace transform

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Problem: Solve the equation: \[ f(t)=\int_{0}^{t} \frac{\operatorname{ch} \tau-1}{\tau} d \tau . \]

10.4.10 Laplace transform

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Problem: Find the Laplace transform \( F(p) \) by the given signal: \[ f(t)=\int_{0}^{t} \frac{\cos \beta \tau-\cos \alpha \tau}{\tau} d \tau . \]

10.4.11 Laplace transform

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