Equation of streamline is given as

$\left( {\frac{{dy}}{{dx}}} \right) = \left( {\frac{v}{u}} \right)$ therefore,

$\left( {\frac{{dy}}{{dx}}} \right) = \frac{{\frac{{4y}}{{\left( {{x^2} + {y^2}} \right)}}}}{{\frac{{4x}}{{\left( {{x^2} + {y^2}} \right)}}}}$

$\left( {\frac{{dy}}{{dx}}} \right) = \frac{{4y}}{{\left( {{x^2} + {y^2}} \right)}} \times \frac{{\left( {{x^2} + {y^2}} \right)}}{{4x}}$

$= \left( {\frac{y}{x}} \right)$

$\frac{{dy}}{{dx}} = \frac{y}{x}$

$\frac{{dy}}{y} = \frac{{dx}}{x}$

$\ln y = \ln x + c$

$\ln y – \ln x = c$

$\ln \left( {\frac{y}{x}} \right) = \ln c$

$\frac{y}{x} = \ln c$

$y = {c_1}x$

The streamlines are straight lines from a source.