I've been tasked with the following. Consider $x^\prime(t)=(x^2(t).t^2)/(tx(t))$ with $x(1)=1$.
a) Show the IVP is homogeneous.
b)Find x: I $\longrightarrow$ $\mathbb{R}$ specifying the maximal interval I where x is defined and solves the equation.
Here's how I started:
I started by defining a homogeneous equation $f:G \longrightarrow$ $\mathbb{R}$ i.e. for λ in $\mathbb{R}$ \{0}: $1. (λt, λx)$ is in $G$ and $2. f(λt,xt)=f(t,x)$ for $(t,x) \in G$.
For $1$, I came to the conclusion that our $G$ is $\mathbb{R}$ \{0} hence for our defined lambda $(λt,λx) \in \mathbb{R}$ \{0} or $G$.
As for two I'm not too sure how to continue as I'm not sure how I can link my given definition for a homogeneous equation to a homogeneous IVP. Do I need to show $λx'(λt)=x'(t)$? I'm not too sure. p.s. apologies for my poor use of mathjax, I'm quite new to it.