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I want to find a scalar potential $\varphi$ for the vectror field

$$\mathbf{F}(x,y) = (2\cdot x \cdot y +x )\mathbf{i}+{x^2}\mathbf{j}$$

such that $\varphi(0,0)=5$

First I need to check that vector field is conservative

$$\frac{\delta F_1}{\delta y}=\frac{\delta F_2}{\delta x}$$

$$2x=2x \to \frac{\delta F_1}{\delta y}-\frac{\delta F_2}{\delta x} = 0$$ so the vector field is conservative.

Then I want to find the scalar potential. I know that $\nabla f= \mathbf F $

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Form the first component we have

$$f_x=2\cdot x \cdot y +x \implies f=x^2y+\frac12x^2+g(y)$$

and from the second one

$$f_y=x^2+g'(y)=x^2 \implies g(y)=k$$

therefore

$$f(x,y)=x^2y+\frac12x^2+k$$

with constant $k$ to be determined by the given condition.

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