MTH423
Question: Which of the following function is a solution of the Volterra type integral equation \\( f(x)=x+ \\int_{0}^{x} sin (x-t) f(t)dt\\) ?
Answer: \\(x- \\frac{x^3}{6}\\)
Question: There is a continuous curve which does not have a point of self-intersection it is said as?
Answer: Simple curve
Question: What is the resolvent kernel for the integral equation \\(u(x) = F(x) + \\int_{logâ¡2}^x u(t)dt.\\)
Answer: 1
Question: Physially, integrating \\(\\int_{a}^{b} f(x)dx\\) means finding the
Answer: area under the curve from a to b
Question: Write out the initial value problem corresponding to the integral equation \\(y(x) = 1+ \\int_{0}^{x} y(t)dt\\)
Answer: \\(y^\’-y=0, y(0)=1\\)
Question: Find the eigen value, λ of this Fredholm integral equation \\( y(x)=λ \\int_{0}^{1} x^2 t y(t)dt \\)
Answer: 4
Question: The mean value of a function \\( f(x)\\) from a to b is given by any of the following:
Answer: \\(\\frac{\\int_{a}^{b} f(x)}dx}{b-a}\\)
Question: What is the value of \\(α \\)for which the integral equation \\( u(x)= α \\int_{0}^{1 }e^{x-t} u(t)dx\\) has a non – trivial solution.
Answer: 1
Question: For a continuous function \\(f(t)\\), \\(0≤t ≤1\\), the integral equation \\(y(t) = f(t)+3 \\int_{0}^{1} t s y(s)ds\\) has
Answer: infinity many solutions, if\\( \\int{0}^{1) s f(s)ds ≠0\\)
Question: The exact value of \\(\\int_{0.2}^{2.2} x e^x dx\\) is most nearly
Answer: 11.807
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