Problem 03 | Equations with Homogeneous Coefficients

Problem 03
2(2x2+y2)dxxydy=0
 

Solution 03
2(2x2+y2)dxxydy=0
 

Let
y=vx

dy=vdx+xdv
 

2(2x2+v2x2)dxvx2(vdx+xdv)=0

4x2dx+2v2x2dxv2x2dxvx3dv=0

4x2dx+v2x2dxvx3dv=0

x2(4+v2)dxvx3dv=0

x2(4+v2)dxx3(4+v2)vx3dvx3(4+v2)=0

dxxvdv4+v2=0

Problem 02 | Equations with Homogeneous Coefficients

Problem 02
(x2y)dx+(2x+y)dy=0
 

Solution 02
(x2y)dx+(2x+y)dy=0
 

Let
y=vx

dy=vdx+xdv
 

Substitute,
(x2vx)dx+(2x+vx)(vdx+xdv)=0

xdx2vxdx+2vxdx+2x2dv+v2xdx+vx2dv=0

xdx+2x2dv+v2xdx+vx2dv=0

(xdx+v2xdx)+(2x2dv+vx2dv)=0

x(1+v2)dx+x2(2+v)dv=0

Problem 01 | Equations with Homogeneous Coefficients

Problem 01
3(3x2+y2)dx2xydy=0
 

Solution 01
3(3x2+y2)dx2xydy=0
 

Let
y=vx

dy=vdx+xdv
 

Substitute,
3(3x2+v2x2)dx2vx2(vdx+xdv)=0

3(3+v2)x2dx2vx2(vdx+xdv)=0
 

Divide by x2,
3(3+v2)dx2v(vdx+xdv)=0

9dx+3v2dx2v2dx2vxdv=0

9dx+v2dx2vxdv=0

(9+v2)dx2vxdv=0