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The equation is $$ \Delta u+cu=0 $$

on the $\mathbb{R}^2$ plane, where $c$ is a constant. My purpose is to find a suitable constant to get a solution of this PDE.

My idea is to let $u(x,y)=g(x^2+y^2)$, then the equation turns into the following ODE:

$$ 4rg''(r)+4g'(r)+cg(r)=0 $$

So I set $c=4$ and try to solve the ODE

$$ rg''(r)+g'(r)+g(r)=0 $$

However, I failed to solve it.

Can anyone help me?

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What kind of solution are you looking for? It is possible to find many explicit solutions for any given value of the constant $c$.

  • Solutions that depend only on $x$. Depending on the sign of $c$ this gives $u=A\,\sin(\sqrt{c}\,x)+B\,\cos(\sqrt{c\,}x)$ (if $c>0$), $u=A\,\sinh(\sqrt{-c}\,x)+B\,\cosh(\sqrt{-c\,}x)$ (if $c<0$), $u=A\,x+B$ (if $c=0$).
  • Solutions that depend only on $y$: change $x$ by $y$ in the above.
  • Solutions of the form $X(x)Y(y)$. This leads to $$ -\frac{X''}{X}=\frac{Y''}{Y}+c=\text{constant} $$ Form here you can get solutions like $u=\cos(\lambda\,x)\sin(\sqrt{c-\lambda^2}\,y)$ if $0<\lambda^2<c$.
  • Radial solutions, which leads to Bessel functions.
  • Linear combinations of the above.
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Take $r = t^2$ and look up Bessel's equation.

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Looks like you are trying to solve the homogeneous Helmholtz equation in 2 dimensions. Following the method described here [ ], you will get a solution $$ u(r,\theta)= \sum_{n=1}^\infty J_n( \sqrt{c} r) \left[ c_{n} \sin n\theta + d_{n}\cos n\theta \right] $$

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