Question

Find the linear substitution that reduces the inhomogeneous equation to the homogeneous one: y^,=x+y-3/y-x+1

Answer 1

A first order differential equation

$dy/dx=f(x,y)$

is called homogeneous equation, if the right side satisfies the condition

$f(tx,ty)=f(x,y)$

we have:

$f(x,y)=x+\frac{y-3}{y-x}+1$

then:

$f(tx,ty)=tx+\frac{ty-3}{ty-tx}+1$

$x+\frac{y-3}{y-x}+1=tx+\frac{ty-3}{ty-tx}+1$

$x(y-x)+y-3=t^2x(y-x)+ty-3$

$x(y-x)(t^2-1)+y(t-1)=0$

$x(y-x)(t+1)+y=0$

$t=-\frac{y}{x(y-x)}-1$