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For partial derivatives of a function use the subscript notation; so for the second partial derivative of the function u(x,t) with respect to x use uxx. For ordinary differential equations use the prime notation, so the second derivative of the function f(x) is f''.

Solve the heat equation

$\displaystyle k\frac{\partial^2 u}{\partial x^2} + 13 = \frac{\partial u}{\partial t}, 0< x < 3, t > 0$

$\displaystyle u(0,t) = 86, u(3,t) = 97, t > 0$

$\displaystyle u(x,0) = x\!\left(3-x\right), 0 < x < 3$

using a steady-state and transient solution: ie write $u(x,t) = v(x,t) + S(x)$ with $v$ a solution of the heat equation with homogeneous boundary conditions

$\displaystyle k\frac{\partial^2 v}{\partial x^2}= \frac{\partial v}{\partial t}, 0< x < 3, t > 0$

$\displaystyle v(0,t) = 0, v(3,t) = 0, t > 0$

$\displaystyle v(x,0) = x\!\left(3-x\right)-S(x), 0 < x < 3$

Using the substitution $u = v + S$

$\displaystyle k\frac{\partial^2 u}{\partial x^2}$ =

$\displaystyle\frac{\partial u}{\partial t}$ =

So the steady-state solution S must satisfy the IVP (DE first, then IC's)

$\displaystyle = 0$

$\displaystyle S(0) =$

$\displaystyle S(3) =$

Therefore $S(x) =$
and $v(x,t) = \frac{2}{3}\displaystyle\sum\limits_{n=1}^\infty b_n e^{\left( -k\frac{n^2\pi^2}{{3}^2}t\right)} \sin\left( \frac{n\pi}{3}x\right)$
with
$b_n = \displaystyle\int_0^{3}$ $\hskip 3pt dx$

Therefore the solution of nonhomogeneous heat equation is
$u(x,t) = v + S =$
$\frac{2}{3}\displaystyle\sum\limits_{n=1}^\infty\Bigg($ $e^{\left( -k\frac{n^2\pi^2}{{3}^2}t\right)} \sin\left( \frac{n\pi}{3}x\right)\Bigg) +$

You can earn partial credit on this problem.