How Can I Solve This Partial Differential Equation? Wolfram Alpha Can't Interpret It Right.
$\Begingroup$ I Stumbled Upon a Differential Equation Which I Do Not Know How to Solve but Would Love to Know the Answer. I Tried Plugging It in Wolfram Alpha...
I stumbled upon a differential equation which I do not know how to solve but would love to know the answer. I tried plugging it in wolfram alpha but it didn't help. For some reason WA wasn't interpreting it right.
$$ \frac{ \partial y}{\partial x} \bigg( { \frac{\partial^2 y}{\partial \epsilon \partial x}\bigg) } = 0 $$
I am looking for $y(x, \epsilon)$ with these conditions: $$\frac{\partial y}{\partial x} {(0, \epsilon)} = 0$$ $$y(x, \epsilon_0) = y(x, -\epsilon_0) = h$$ where $h \in \mathbb{R}_{>0}$
If not analytical, can someone at least give me a hint as to what the numerical solution would like so that I know, intuitively, if this model is right.
1 Answer
I have put it in Mathematica:
DSolve[D[y[x, e], x]*D[y[x, e], {x, 1}, {e, 1}]*Sqrt[1 + (D[y[x, e], x])^2] == 0
&& (D[y[x, e], x] /. x -> 0) == 0, y, {x, e}]
and I got a simple linear solution in $\epsilon$
{{y -> Function[{x, e}, C[2][e]]}}
in other words: $y[x,\epsilon]= C(\epsilon)$.
So the derivative of y in x is always zero but the boundary condition equal to h will be satisfied simultaneously only for even function $C(\epsilon)=C(-\epsilon)$.
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