GATE Mechanical Engineering (ME) 2019 Shift 1 Solved Paper

Since the harmonic function is analytic, so it satisfies the Cauchy-Riemann (C-R) equation
Therefore, $\frac{\partial \mathit{u}}{\partial \mathit{x}}=\frac{\partial \mathit{v}}{\partial \mathit{y}}$ and $\frac{\partial \mathit{u}}{\partial \mathit{y}}=-\frac{\partial \mathit{v}}{\partial \mathit{x}}$
Given $\mathit{u}=2{\mathit{x}}^2-2{\mathit{y}}^2+4\mathit{x}\mathit{y}$
$\Rightarrow \frac{\partial \mathit{u}}{\partial \mathit{x}}=4\mathit{x}+4\mathit{y}=\frac{\partial \mathit{v}}{\partial \mathit{y}}$
So, on integrating, $\mathit{v}=4\mathit{x}\mathit{y}+2{\mathit{y}}^2+\mathit{f}\left(\mathit{x}\right)$
Now, $\frac{\partial \mathit{v}}{\partial \mathit{x}}=4\mathit{y}+{\mathit{f}}^{\prime }\left(\mathit{x}\right)=-\frac{\partial \mathit{u}}{\partial \mathit{y}}=4\mathit{y}-4\mathit{x}$
So, ${\mathit{f}}^{\prime }\left(\mathit{x}\right)=-4\mathit{x}$
Or, $\mathit{f}\left(\mathit{x}\right)=-2{\mathit{x}}^2+$ constant
Therefore, $\mathit{v}=4\mathit{x}\mathit{y}+2{\mathit{y}}^2-2{\mathit{x}}^2+$ constant $\mathit{b}$