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Questions and Answers
What are the conditions for the existence of the Laplace transform $G(s)$ for the function $g(x)$?
What are the conditions for the existence of the Laplace transform $G(s)$ for the function $g(x)$?
- $g(x)$ is piecewise continuous on the interval of integration $0 \leq x < B$ for any positive $B$, and $g(x)$ is of exponential order $e^{cx}$ as $x \to \infty$. (correct)
- $g(x)$ is piecewise continuous on the interval of integration $0 \leq x < B$ for any positive $B$, and $g(x)$ is of logarithmic order $\log(x)$ as $x \to \infty$.
- $g(x)$ is piecewise continuous on the interval of integration $0 \leq x < B$ for any positive $B$, and $g(x)$ is of trigonometric order $\sin(\omega x)$ as $x \to \infty$.
- $g(x)$ is piecewise continuous on the interval of integration $0 \leq x < B$ for any positive $B$, and $g(x)$ is of polynomial order $x^n$ as $x \to \infty$.
What is the property of the Laplace transform that states $L{a g(x) + b h(x)} = a L{g(x)} + b L{h(x)}$, where $a$ and $b$ are constants?
What is the property of the Laplace transform that states $L{a g(x) + b h(x)} = a L{g(x)} + b L{h(x)}$, where $a$ and $b$ are constants?
- Multiplication by $x$ property
- Linearity property (correct)
- Constant Multiple property
- Differentiation property
What is the property of the Laplace transform that states $L{x g(x)} = -G'(s)$?
What is the property of the Laplace transform that states $L{x g(x)} = -G'(s)$?
- Multiplication by $x$ property (correct)
- Constant Multiple property
- Differentiation property
- Linearity property
