For a uniform flow velocity potential \(\phi=V_{\infty}x\) and stream function \(\psi=V_{\infty}y\).Laplace equation is given as \[\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}=0\;;\;\frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}=0\]

Velocity potential\[\phi=V_{\infty}x
\\\frac{\partial \phi}{\partial x}=V_{\infty}\;,\;\frac{\partial^2 \phi}{\partial x^2}=0
\\\frac{\partial \phi}{\partial y}=0\;,\;\frac{\partial^2 \phi }{\partial y^2}=0\]

Therefore Laplace equation \[\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}=0+0=0\]

is satisfied.

Stream function \[\psi=V_{\infty}y\]

\[\frac{\partial\psi }{\partial x}=0\;,\;\frac{\partial^2 \psi}{\partial x^2}=0
\\\frac{\partial \psi}{\partial y}=V_{\infty}\;,\;\frac{\partial^2 \psi}{\partial y^2}=0\]

Therefore Laplace equation \[\frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}=0+0=0\]

is satisfied.