Practice Step-by-step Solution Using Laplace Transforms (18.2) - Application to Integral Equations

Practice: Step-by-Step Solution Using Laplace Transforms

Lesson 18.2Application to Integral Equations

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8 questions

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6 questions · about 4 min · pass mark 70%. Hints available.
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2 questions · about 1 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
Define a Volterra Integral Equation.
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Look for the structure of the integral involving an unknown function.
Question 2.
What does the kernel represent in an integral equation?
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Consider it as a weight assigned to different inputs.
3Question 3.
What type of equation is $f(t) = g(t) + \int_0^t K(t-\tau) f(\tau) d\tau$ ?
- Differential Equation
- Integral Equation
- Algebraic Equation
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Look for the presence of an integral with an unknown.
4Question 4.
True or False: The Convolution Theorem allows the Laplace Transform of a convolution to be expressed as a product.
- True
- False
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Think about how operations change during transformation.
Question 5.
Prove that if $f(t) = \int_0^t f(\tau) d\tau$ , the solution can be expressed as $f(t) = te^t$ using Laplace Transforms.
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Focus on how integral arguments are expressed in the transformed domain.
Question 6.
Given $K(t - \tau) = sin(t - \tau)$ , derive the explicit formula for $f(t)$ if $g(t) = t^2$ .
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You will need to remember properties of sine when finding inverse transforms.

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