Differentials — Question 7

Question 7 – Solution

Strategy. If $df=M\,dx+N\,dy$, then $M_y=N_x$ on a simply connected domain.

Step 1: Compatibility Here $$M=2xy+3,\qquad N=x^2+4y.$$ Then $$M_y=2x=N_x,$$ so the form is compatible throughout $\mathbb R^2$.

Step 2: Recover $f$ Integrate $f_x=M$ in $x$: $$f=x^2y+3x+C(y).$$ Then $$f_y=x^2+C'(y)=x^2+4y,$$ so $C'(y)=4y$ and $C(y)=2y^2+C_0$.

Result and verification $$\boxed{f(x,y)=x^2y+3x+2y^2+C_0}.$$ Indeed, differentiating gives $df=(2xy+3)\,dx+(x^2+4y)\,dy$. The additive constant is invisible to the differential.