Homogeneous ODE definition

Question

I'm taking a class on Ordinary Differential Equations and I saw two definitions of a homogeneous first order ODE $x'(t)=f(t,x(t))$:

(i) $f$ is homogeneous of degree $0$ if for every $c\in\mathbb{R}$ the function $f$ satisfies $f(ct,cx)=f(t,x)$,

(ii) $f$ is homogeneous if $f$ only depends of $\frac{x}{t}$, i.e., there is a function $F$ such that $f(t,x)=F(\frac{x}{t})$.

How can I show that the definition (i) implies the definition (ii)? I think this might be a little bit obvious but I can't see it.

Thanks for any help.

Answer 1

$$x'(t)=f(t,x(t))$$ Substitute : $$x=zt \implies x'=z't+z$$ $$z't+z=f(t,zt)$$ $$z't+z=f(1,z)$$ $$ \implies z't+z=F(z)=F\left (\dfrac xt \right)$$

Answer 2

Take $(t_0,x_0)$ with $t_0 \neq 0$ and define

$$F(y) = f(t_0, t_0 y).$$

For any $c \in \mathbb R$ you get $f(ct,cx)=F(x/t)=f(t_0,t_0x/t) = f(t,x)$