Диференциално уравнение

[Гост]

(10x4+y3)dx+3y(xy-2)dy=0

[Добромир Глухаров]

[tex]\frac{\partial U}{\partial x}=P(x,y)=10x^4+y^3=U_x(x,y)[/tex]
[tex]\frac{\partial U}{\partial y}=Q(x,y)=3y(xy-2)=3xy^2-6y=U_y(x,y)[/tex]

[tex]U_{xy}(x,y)=P_y=Q_x=3y^2[/tex] - уравнението е точно

[tex]U(x,y)=\int(10x^4+y^3)dx+\varphi(y)=2x^5+xy^3+\varphi(y)[/tex]

[tex]U_y=3xy^2+\varphi'(y)=3xy^2-6y[/tex]

[tex]\varphi(y)=\int(-6y)dy=C_1-3y^2[/tex]

[tex]U(x,y)=2x^5+xy^3-3y^2+C_1=C_2[/tex]

[tex]2x^5+xy^3-3y^2=C[/tex]