I.7 verify that the given function is a solution

[karush]

$\textsf{i.7 verify that the given function is a solution} $
\begin{align*}\displaystyle
u_{xx}+u_{yy}+u_{zz}&=0\\
u=\phi(x,y,z)&=(x^2 + y^2 +z^2)^{-1/2}\\
(x,y,z)&\ne(0,0,0)
\end{align*}

ok there is no book answer
$\tiny{Elementary \, Differential \, Equations \, Boundary \, Value \, Problems \quad Boyce/Diprinaia \quad 1965}$

 

[Prove It]

$\textsf{i.7 verify that the given function is a solution} $
\begin{align*}\displaystyle
u_{xx}+u_{yy}+u_{zz}&=0\\
u=\phi(x,y,z)&=(x^2 + y^2 +z^2)^{-1/2}\\
(x,y,z)&\ne(0,0,0)
\end{align*}

ok there is no book answer
$\tiny{Elementary \, Differential \, Equations \, Boundary \, Value \, Problems \quad Boyce/Diprinaia \quad 1965}$

So evaluate those derivatives and see if they combine in that way to give 0...

 

[karush]

So evaluate those derivatives and see if they combine in that way to give 0...

ok I am like page 5 in the book so I don't know exactly what $U_{xx}$ means I presume it means 2nd derivative of the function U

 

[MarkFL]

That notation usually means the second partial derivative w.r.t $x$.

 

[karush]

so what would we use for a function?

 

[MarkFL]

so what would we use for a function?

\(\displaystyle u=\phi(x,y,z)=\left(x^2 + y^2 +z^2\right)^{-\Large\frac{1}{2}}\)