Reducing a Differential Equation of a Special Form to a Homogeneous Equation

coefficients of numerator

1

1

2

2

1

1

coefficients of denominator

1

1

1

1

′

y

F

x+2y+1

x+y+1

The system of linear equations

x+2y+10

x+y+10

has a unique solution

x-1

y0.

Substitute:

Xx+1, xX-1

Yy, yY

Xx

Yy

x+2y+1X+2Y

x+y+1X+Y

to get

Y

X

F

X+2Y

X+Y

This is a homogeneous

differential equation.

This Demonstration shows the reduction of a differential equation of the form

y'=F((ax+by+c)/(dx+ey+f))

to a homogeneous differential equation of the form

Y'=F((aX+bY)/(dX+eY))

. This case occurs if the system of linear equations

ax+by+c=0

,

dx+ey+f=0

has a unique solution

x

1

,

y

1

; then new variables are introduced by the equations

X=x-

x

1

,

Y=y-

y

1

. If the system of linear equations has no solution or has infinitely many solutions, the differential equation reduces to an equation with separable variables.