Practice Example Problems (18.3) - Application to Integral Equations - Mathematics - iii (Differential Calculus) - Vol 1

Practice: Example Problems

Lesson 18.3Application to Integral Equations

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10 questions

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6 sample questions

Question 1.
What is the general form of a Volterra integral equation of the second kind?
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Think about the elements involved in the equation structure.
Question 2.
What does the kernel $K(t - \tau)$ represent?
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Consider what role it plays in the behavior of the integral.
3Question 3.
What is an integral equation?
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Think about the general definition of integral equations.
4Question 4.
Does the kernel in a Volterra equation affect the solution?
- True
- False
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Consider the effect of the kernel on previous function values.
Question 5.
Use Laplace Transforms to solve the Volterra integral equation: $f(t) = \cos(t) + \int_{0}^{t} \sin(t - \tau)f(\tau)d\tau$ .
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Focus on identifying the correct kernel and solving dimensions carefully.
Question 6.
Derive the solution for the integral equation: $f(t) = 2 + \int_{0}^{t} (t - \tau)^2 f(\tau) d\tau$ using transformations.
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Make sure to handle the squaring in the kernel when applying transformations.

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