∂f∂x=2x+y2\frac{\partial f}{\partial x} = 2x + \frac{y}{2}∂x∂f=2x+2y ∂f∂y=ax+y\frac{\partial f}{\partial y} = ax + y∂y∂f=ax+y ∫f(x) dx=x2+y2x+C1∫f(y) dy=axy+12y2+C2\int f_{(x)}\,dx = x^2 + \frac{y}{2}x + C_1 \qquad \int f_{(y)}\,dy = axy + \frac{1}{2}y^2 + C_2∫f(x)dx=x2+2yx+C1∫f(y)dy=axy+21y2+C2 12xy=axya=12\frac{1}{2}xy = axy \qquad a = \frac{1}{2}21xy=axya=21 よって f(x,y)=x2+12xy+12y2+C(C は積分定数)f_{(x,y)} = x^2 + \frac{1}{2}xy + \frac{1}{2}y^2 + C \qquad (\text{C \text{は積分定数}})f(x,y)=x2+21xy+21y2+C(C は積分定数)