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WOLFRAM|DEMONSTRATIONS PROJECT

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
1
1
y
F
x+2y+1
x+y+1
The system of linear equations
x+2y+10
x+y+10
has a unique solution
x-1
y0.
Substitute:
Xx+1, xX-1
Yy, yY
Xx
Yy
x+2y+1X+2Y
x+y+1X+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.
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