Problem # 1 Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form: Insulated 0° kuxx-hu= ut, 0 0 Insulated L 0° Heat transfer from lateral surface of the rod X Find the temperature u(x, t) if the initial temperature is f(x) throughout and the ends x = 0 and x = L are insulated.

Q: 1) (i) Find the gradient of f. (ii) Evaluate the gradient at the point P. f(r, y) = 12y-3 sin(y), P…

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Q: know that a3 .8

A: Antiderivative: F(x)=f(x9) F'(x)=f'(x9)*ddx(x9)       =f'(x9)*9x8 F'(a)=9f'(a9)a8       =9*13*6…

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A: Slopes of y's yy'=y(2-y)y=3-3y=20y=11y=00

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Q: 4. du =u -1 dt du =v -1

A: The system is dudt=u-1 and dvdt=v-1. The vector represented by it is u-1 , v-1 Take I quadrant: At…