Partial Differential Equations : Heat Equations
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1) Let A(x,y) be the area of a rectangle not degenerated of dimensions x and y, in a way that the rectangle is inside a circle of a radius of 10. Determine the domain and the range of this function.
2) The wave equation (c^2 ∂^2 u / ∂ x^2 = ∂^2 u / ∂ t^2) and the heat equation (c ∂^2 u / ∂ x^2 = ∂ u / ∂ t) are two of the most important equations of physics (c is a constant). They are called partial differential equations. Show the following:
a) u = cos x cos ct and u = e^x cosh ct satisfies the wave equation.
b) u = e^-ct sin x and u = t^-1/2 e^[(-x^2)/(4ct)] satisfies the heat equation.
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Partial differential equations problems
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I need the solution for this problems and the steps of how it was done, even if its handwritten.
1) Let A(x,y) be the area of a rectangle not degenerated of dimensions x and y, in a way that the rectangle is inside a circle of a radius of 10. Determine the domain and the range of this function.
We can put the center of the rectangle at the same center of the circle. See the graph above. We can set up an inequality as follows. Assume the dimensions of this rectangle are x and y. Hence,
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Education
- BSc , Wuhan Univ. China
- MA, Shandong Univ.
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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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