If an incompressible fluid flows in a corner bounded by walls meeting at the origin at an angle of 60', the streamlines of the flow satisfy the equation $\mathbf{2}\mathit{x}\mathit{y}\mathit{d}\mathit{x}\mathbf{+}\left(x^2-y^2\right)\mathit{d}\mathit{y}\mathbf{=}\mathbf{0}$. Find the streamlines.

The given equation is$2xydx+\left(x^2-y^2\right)dy=0$.

A first-order differential equation (of one variable) is called exact, or an exact differential, if it is the result of a simple differentiation. The equation $\mathit{P}\left(x,y\right)\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathit{Q}\left(x,y\right)\mathbf{=}\mathbf{0}$, or in the equivalent alternate notation as $\mathit{P}\left(x,y\right)\mathit{d}\mathit{y}\mathbf{+}\mathit{Q}\left(x,y\right)\mathit{d}\mathit{x}\mathbf{=}\mathbf{0}$, is exact if ${\mathit{P}}_{\mathbf{x}}\left(x,y\right)\mathbf{=}{\mathit{Q}}_{\mathbf{y}}\left(x,y\right)$.

The given equation is $2xydx+\left(x^2-y^2\right)dy=0.....\left(1\right)$

Let,

$$\begin{gathered} M=2xy \\ N=\left(x^2-y^2\right) \end{gathered}$$

Differentiate M with respect to y, and N with respect to x as,

$$\begin{gathered} M_y=2x \\ {N}_x=2x \end{gathered}$$

It can be noticed that $M_y=N_x$

Therefore, the equation is exact differential equation.

The solution of the exact differential equation is,

$F\left(x,y\right)=$Constant

Now,

$$\begin{gathered} F_x=\int 2xydx \\ =\frac{2x^{2}y}{2}+c \\ =x^{2}y+c_1 \end{gathered}$$

And,

$$\begin{gathered} F_y=\int \left(x^2-y^2\right)dy \\ =x^{2}y-\frac{y^3}{3}+c_2 \end{gathered}$$

Now, for $F_x=F_y$

$c_2=0\&c_1=-\frac{y^3}{3}$

Therefore, the equation of the streamlines is $F\left(x,y\right)=x^{2}y-\frac{y^3}{3}$.