homogeneous equations

Homogeneous Equation :
The first-order differential eq .
dy/dx=f(x,y)
is homogeneous differential eq.
If having the property that f(tx,ty)=f(x,y)
For every real number t .
Example : 1. The equation . y^ =(y+x)/x is homogeneous , since
f(tx,ty)=(TY+TX)/TX=(T(Y+X))/TX=(Y+X)/X=f(x,y)
2.The equation y^ =y^2/x is not homogeneous since
f(tx,ty)=(ty)^2/tx=(t^2 y^2)/tx=t y^2/x?f(x,y)
The homogeneous eq. can be solved by reduction into separable eq. by substitution y=xv or v=y/x
With its corresponding derivative dy/dx=v+x dv/dx
The resulted eq. is on the variable v and solved as separable eq. and then back substitution on v
Example : To solve the homogeneous eq. ; y^ =(2xy )/(x^2-y^2 ) substitute y=xv in the eq. then ;
v+x dv/dx=2x(xv)/(x^2-(xv)^2 )
Simplified as ;
x dv/dx=-(v(v^2+1))/(v^2-1)
Or
1/x dx+((v^2-1))/(v(v^2+1)) dv=0
Using partial functions as
1/x dx+(-1/v+2v/(v^2+1))dv=0
By integrating both sides , we obtain
ln|x|-ln|v|+ln(v^2+1)=c
Can simplified to
x(v^2+1)=kv for c=ln|k|
Substitute v=y/x we find the solution as ; x^2+y^2=ky
Exercises :
Q1: Write the differential equations in standard and differential forms
y(yy^ -1)=x
dy/dx=y/x
Q2: Show which equations is homogeneous and then solve it ;
y^ =(x^2+y)/( x^3 )
y^ =(2xy e^(x/y))/(x^2+y^2 sin x/y)
y^ =(x^2+y^2 )/xy