Determine all product solutions for $\frac{\partial u^2 }{\partial x^2}=\frac{\partial u^2 }{\partial y^2}$.
Decide one solution that satisfies $\left\{\begin{matrix} f(0)=0\\f(1)=0 \\f(0.25)=0.5 \\g(0.25)=g(0.50)=1 \end{matrix}\right.$
I shall also determine the separation constant that meets the conditions.
I've done: $$u(x,y)=X(x)Y(y)\\ \frac{\partial u^2 }{\partial x^2}=X''Y,\ \frac{\partial u^2 }{\partial y^2}=Y''X, \frac{X''}{X}=\frac{Y''}{Y}=-k\ (constant)\\$$
If $k=0$: $u(x,y)=(c_1x+c_2)(c_3y+c_4)$
If $k<0, k=-a^2$: $u(x,y)=(c_1e^{ax}+c_2e^{-ax})(c_3e^{ay}+c_4e^{-ay})$
If $k>0, k=a^2$: $u(x,y)=(c_1cos(ax)+c_2sin(ax))(c_3cos(ay)+c_4sin(ay))$
From $u(x,y)=(c_1x+c_2)(c_3y+c_4)$, I get $f(x)=(c_1x+c_2)$ and $g(y)=(c_3y+c_4)$.
$$f(0)=0\Rightarrow 0=c_2\\ f(1)=0\Rightarrow c_1=-c_2\\ f(0.25)=0.5\Rightarrow c_1=2-4c_2\\ -c_2=2-4c_2\Rightarrow c_2=\frac{2}{3}\Rightarrow c_1=-\frac{2}{3}\\ c_2=0\Rightarrow c_1=0\Rightarrow 0=2-4\cdot 0\Rightarrow 2=0$$
But that doesn't add up. I'm obviously missing something and making mistakes. What is it?
EDIT: Updated some spelling errors.
EDIT 2: From $u(x,y)=(c_1e^{ax}+c_2e^{-ax})(c_3e^{ay}+c_4e^{-ay})$, I get $f(x)=c_1e^{ax}+c_2e^{-ax}$ and $g(y)=c_3e^{ay}+c_4e^{-ay}$.
$$f(0)=0 \Rightarrow 0=c_1+c_2 \Rightarrow c_1=-c_2\\ f(1)=0 \Rightarrow 0=c_1e^a-c_1e^{-a} \Rightarrow 0=e^a-e^{-a} \Rightarrow 0=a+a \Rightarrow a=0\\ f(0.25)=0.5 \Rightarrow 0.5=c_1-c_1 \Rightarrow 0.5= 0$$
From $u(x,y)=(c_1cos(ax)+c_2sin(ax))(c_3cos(ay)+c_4sin(ay))$, I get $f(x)=c_1cos(ax)+c_2sin(ax)$ and $g(y)=c_3cos(ay)+c_4sin(ay)$.
$$f(0)=0 \Rightarrow 0=c_1\\ f(1)=0 \Rightarrow 0=c_2sin(a) \Rightarrow 0=a\\ f(0.25)=0.5 \Rightarrow 0.5=c_2sin(0) \Rightarrow 0.5=0$$
I'm ending up with $2=0$, $0.5=0$ and $0.5=0$. What am I missing?
EDIT 3: From $u(x,y)=(c_1cos(ax)+c_2sin(ax))(c_3cos(ay)+c_4sin(ay))$, I get $f(x)=c_1cos(ax)+c_2sin(ax)$ and $g(y)=c_3cos(ay)+c_4sin(ay)$.
$$f(0)=0 \Rightarrow 0=c_1\\ f(1)=0 \Rightarrow 0=c_2sin(a) \Rightarrow a=2\pi n \\ f(\frac{1}{4})=\frac{1}{2} \Rightarrow \frac{1}{2}=c_2sin(\frac{2\pi}{4}) \Rightarrow \frac{1}{2}=c_2\\ g(\frac{1}{4})=1 \Rightarrow 1=c_3cos(\frac{2\pi }{4})+c_4sin(\frac{2\pi }{4})\Rightarrow 1=c_4\\ g(\frac{1}{2})=1 \Rightarrow 1=c_3cos(\frac{2\pi }{2})+c_4sin(\frac{2\pi }{2})\Rightarrow -1=c_3\\ \\ u(x,y)=(c_1cos(ax)+c_2sin(ax))(c_3cos(ay)+c_4sin(ay))\\ \Rightarrow u(x,y)=(\frac{1}{2}sin(2\pi x))(-cos(2\pi y+sin(2\pi y)))$$
Separation constant: $$a=2\pi \\ k=a^2 \\ k=(2\pi)^2=4\pi^2$$