Answer to Question #287261 in Civil and Environmental Engineering for apple

Question

solution of the Differential equation

7. (1+y² + xy²)dx + (x²y + y + 2xy)dy =0

8. 2ydx + x(x² Iny-1)dy = 0

9. y(2x+y²)dx + x(y²-x)dy = 0

10.(3x²-2y2)y' = 2xy; x = 1, y = 1

11.(2x-3y)dx + (2y-3x)dy; x = 1, y = 1

Answer 1

The first consideration is to determine if it is an exact DE.

d_/dy_ (1+y^2 +xy^2) = 2y + 2xy

d_/dx_ (x^2*y + y + 2xy) = 2xy + 2y. The two are equal, therefore, the Differential equation is exact.

The solution (known as the potential function) F(x,y) is derived from

F(x,y) = INTEG {1 + y^2 + xy^2 } dx + g(y) = x + x*y^2 + x^2* y^2 /2 + g(y) and

F(x,y) = INTEG{ x^2* y + y + 2xy } dy = x^ 2* y^2 /2 + y^2/ 2 + x*y^2 + h(x).

On comparison of the two, g(y) = y^2/ 2 and h(x) = x.

Therefore, the solution is :

F(x,y) = C →

x^ 2* y^2 /2 + y^2/ 2 + x*y^2 + x = C