Answer to Question #340898 in Differential Equations for Aiko

Question

Find the general solution of the DE 3y( x2 - 1) dx  (x3  8y – 3x) dy = 0

Answer 1

3y(x²-1)dx+(x³+8y-3x)dy=0

M(x,y)=3y(x²-1), N(x,y)=x³+8y-3x

dM/dy=3x²-3, dN/dx=3x²-3

dM/dy=dN/dx

dF(x,y)=0

F(x,y)=C

F(x,y)=$\intop$ 3y(x²-1)dx+g(y)=3yx³/3-3xy+g(y)=yx³-3xy+g(y)

F'_y=N(x,y)

x³-3x+g'(y)=x³+8y-3x

g'(y)=8y

g(y)=$\int$ 8ydy=4y²+C

Answer: x³y-3xy+4y²=C