Please help me about differential equation?

Question

please kindly help me....any method can be used.....i just dont get the right answer...please help me....

1.(3x-2y+1)dx+(3x-2y+3)dy=0
ans. 5(x+y+c)=2 ln(15x-10y+11)

2.dy/dx =(9x+4y+1)^2
ans. 3 tan(6x+c)=2(9x+4y+1)

3.(3siny-5x)dx+2x^2cotydy=0
ans. x^3(sin y-x)^2=2y+c

4.(x^2+6y^2)dx-4xydy=0,x=1,y=1
ans.2y^2=x^2(3x-1)


please help me....thank you very much !

Answer 1

1. Let u = 3x-2y+3; then du = 3dx - 2dy, so dy = (3 dx - du)/2
Note that the coefficient for dx in the original DE becomes (u-2)

2. Similar to 1. Let u = 9x+4y+1, so du/dx = 9 + 4dy/dx. Then dy/dx = (du/dx - 9)/4

4. We have x² dx + 6y²dy - 4xydy = 0. Look for an integrating factor µ(x), so that ∂(6µy²)/∂y = ∂(-4µxy)/∂x.

I'm still thinking about #3...