separable differential equations? Please help?

Question

solve the following separable equation

1)dy/dx=2/3y^2x^1/3

2)dy/dx=y/(x(x+1)

3)dy/dx=y^1/2xe^x^2

would appreciate the help ... I have the solutions .. however I think I did them all wrong .... thxs

Answer 1

dy/y^2 = 2/3x^1/3 dx
-1/y =1/2 x^4/3 +C so y = -1/(1/2x^4/3+C)
dy/y = dx/x(x+1)dx = 1/x dx -1/(x+1)dx
ln I y I =ln IxI -lnIx+1I+c
I y I = KI x/(x+1I

dy/y^1/2 = x e^x2 dx ( I suppose???)
2y^1/2 = Int e^x^2*x dx
call e^x^2 = z so e^x^2*2xdx =dz and the integral becomes
Int dz /2= z/2+C
2y^1/2 = e^x^2/2 +C

Answer 2

a million) dy/dx - 5y = 0 => dy/dx = 5y => dy/y = 5dx combine the two facets, => ln y = 5x + c you additionally can take consistent as ln c and then take the words with log at the same time. relies upon on the way you like. 2) dy/dx - 2xy = 2x => dy/dx = 2x+2xy => dy/dx = 2x(y+a million) => dy/(y+a million) = 2xdx combine the two facets, => ln(y+a million) = x² + c