wwwwwww
## ZAD 1 i

$LaTe\chi$

$u(a,y) = 0$
$u(x,b) = 0$
$u(0,y) = Asin(\frac{\pi y}{b})$
$u(x,0) = Bsin(\frac{\pi x}{a})$

$u(x,y) = X(x)Y(y)$

$u_{xx}+u_{yy} = 0$

$Y(y)X''(x) + X(x)Y''(y)=0$

$Y(y)X''(x) = - X(x)Y''(y)$

$X''(x)/X(x) = - Y''(y)/Y(y) = \lambda$

$Y'' = -\lambda Y$
$Y'' + \lambda Y = 0$

gdy $\lambda > 0$
$Y(y) = C_1\sin(\sqrt{\lambda}y) + C_2\cos(\sqrt{\lambda}y)$
gdy $\lambda < 0$
$Y(y) = C_1e^{\sqrt{-\lambda}y} + C_2e^{-\sqrt{-\lambda}y}$
gdy $\lambda = 0$
$Y(y) = C_1y + C_2$

gdy $\lambda < 0$
$X(x) = C_1\sin(\sqrt{-\lambda}x) + C_2\cos(\sqrt{-\lambda}x)$
gdy $\lambda > 0$

$X(x) = C_1e^{\sqrt{\lambda}x} + C_2e^{-\sqrt{\lambda}x}$
gdy $\lambda = 0$
$X(y) = C_1x + C_2$

gdy $\lambda < 0$
$X(x)Y(y) = (C_1e^{\sqrt{-\lambda}y} + C_2e^{-\sqrt{-\lambda}y})(C_3\sin(\sqrt{-\lambda}x) + C_4\cos(\sqrt{-\lambda}x))$

gdy $\lambda = 0$
$X(x)Y(y) = (C_1 y + C_2)(C_3x + C_4)$

gdy $\lambda > 0$
$X(x)Y(y) = [C_1\sin(\sqrt{\lambda}y) + C_2\cos(\sqrt{\lambda}y ) ] [C_3e^{\sqrt{\lambda}x} + C_4e^{-\sqrt{\lambda}x}]$

0 = X(a)Y(y)
0 = X(x)Y(b)

Bsin(\pi x / a) = X(x)Y(0)

$\lambda = 0$
$XY = d_1 xy + d_2x+d_3y+d4$

niech $\lambda > 0$
Niech $Y(y) = A\sin(\sqrt{\lambda}y)$
$\lambda = (\frac{\pi}{b})^2$

$X(x) = Ce^{\sqrt{\lambda}x} + De^{-\sqrt{\lambda}x}$
$X(a) = Ce^{\sqrt{\lambda}a} + De^{-\sqrt{\lambda}a} = 0$
$X(0) = 1$

Teraz:
$u(x,b) = 0$, Bo Y(b) = 0
$u(a,y) = 0$, Bo X(a) = 0
$u(x,0) = 0$, Bo Y(0) = 0
$u(0,y) = Asin(\frac{\pi y}{b})$

To potem:
$u(0,b) = 0$
$u(a,0) = 0$
$u(x,0) = Bsin(\frac{\pi x}{a})$
$u(0,y) = 0$

## ZAD 7

$u(x,y) = X(x)Y(y)$

$u_{xx}+u_{yy} = -\lambda u(x,y)$

$Y(y)X''(x) + X(x)Y''(y)= \lambda X(x)Y(y)$

$Y(y)X''(x) = \lambda X(x)Y(y) - X(x)Y''(y)$

$Y(y)X''(x) = X(x)(\lambda Y(y) - Y''(y))$

$\frac{X''(x)}{\lambda Y(y) - Y''(y)} = \frac{X(x)}{Y(y)}$

$\frac{X''(x)}{X(x)} = \frac{\lambda Y(y) - Y''(y)}{Y(y)} = \delta$

$\lambda Y(y) - Y''(y) = \delta Y(y)$
$(\lambda - \delta)Y(y) = Y''(y)$
$Y(y)=C_1e^{\sqrt{(\lambda - \delta)}} + C_2 e^{-\sqrt{(\lambda - \delta)}}$

$X''(x) = \delta X(x)$
$X''(x) - \delta X(x) = 0$
$\delta < 0$
$X_n(x) = C_nsin(\sqrt{-\delta_n}x)$
$\delta_n = -(\frac{n \pi}{a})^2$
$Y''(y) - (\lambda - \delta)Y(y) = 0$
$\lambda_n = \delta_n - (\frac{n \pi}{b})^2$
$Y_n(y) = sin(\sqrt{-\lambda_n + \delta_n}y)$

$sin(\delta x)sin(\delta y)$
sin(\sqrt{-d_n}x)

sqrt(-d_n)cos(sqrt(-d_n)x)
d_n sin(sqrt(-d_n)x)

$\delta_n sin(\sqrt{-\delta_n}x)sin(\sqrt{-\lambda_n + \delta_n}y) + (-\lambda_n + \delta_n)sin(\sqrt{-\delta_n}x)sin(\sqrt{-\lambda_n + \delta_n}y) =$