Question

Solve d^2u/dx^2 + d^2u/dy^2 =0
Y = 0 , Y = 10 , X = infinity
u (x,y) = x - x ^2 at x = 10

Answer 1

Answer on Question #82566 - Math - Differential Equations

Question

Solve

$$\frac {d ^ {2} u}{d x ^ {2}} + \frac {d ^ {2} u}{d x ^ {2}} = 0$$ $$y = 0, y = 10, x = \text{infinity}$$ $$u(x, y) = x - x^{2} \text{ at } x = 10$$

Solution

$$u = X(x)Y(y)$$ $$Y X'' + X Y'' = 0$$ $$\frac {X''}{X} = - \frac {Y''}{Y} = \omega^{2}$$ $$\frac {d^{2} X}{d x^{2}} = \omega^{2} X; \quad \frac {d^{2} Y}{d y^{2}} = - \omega^{2} Y$$

The general solution is:

$$u_{k}(x, y) = X_{k}(x)Y_{k}(y) = \exp \left(- \frac {\omega_{k} x}{10}\right) \sin \frac {\omega_{k} y}{10},$$ $$u(x, y) = \sum_{k=1}^{\infty} a_{k} X_{k}(x)Y_{k}(y) = \sum_{k=1}^{\infty} a_{k} \exp \left(- \frac {\omega_{k} x}{10}\right) \sin \frac {\omega_{k} y}{10}; \quad \omega_{k} = k\pi$$

We have:

$$u(10, y) = \sum_{k=1}^{\infty} a_{k} e^{-\omega_{k}} \sin \frac {\omega_{k} y}{10} = 10 - 100 = -90$$ $$a_{k} = -90 \cdot \frac{2}{10} \int_{0}^{10} e^{-\omega_{k}} \sin \frac {\omega_{k} y}{10} \, dy = 18 e^{-\omega_{k}} \cdot \frac{10}{\omega_{k}} (\cos \omega_{k} - 1) = \frac{180 e^{-\omega_{k}}}{\omega_{k}} ((-1)^{k} - 1)$$

Answer: $u(x,y) = 180\sum_{k=1}^{\infty}((-1)^{k} - 1)\frac{e^{-\omega_{k}}}{\omega_{k}} \exp \left(-\frac{\omega_{k}x}{10}\right) \sin \frac{\omega_{k}y}{10}$.

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