Examples from Existence, uniqueness and linear operators

Reading the hypotheses

The theorem explains the four outcomes from ODE fundamentals:

• For $f(x,y)=x$, the function is continuous and independent of $y$, so the IVP $y'=x$, $y(0)=1$ is uniquely determined.
• For $f(y)=3y^{2/3}$ (interpreting the real power as $|y|^{2/3}$), the Lipschitz condition fails near $y=0$. This failure alone does not imply nonuniqueness, but here the IVP $y'=3y^{2/3}$, $y(0)=0$ has the distinct solutions $y=0$ and $y=x^3$, since $(x^3)'=3x^2=3|x^3|^{2/3}$.
• In $xy'=y-1$, the coefficient of $y'$ vanishes at $x=0$, so the equation cannot be put into the usual normal form $y'=f(x,y)$ at the initial point.
• The equation $|y'|+|y|=0$ forces $y=0$, which is incompatible with $y(0)=1$; no solution satisfies that data.

A particular solution translated by the kernel

For

on an interval with $x\neq0$, take

The homogeneous equation $T[y]=0$ has solution

A particular solution is $y_p=x^2/3$, since

Therefore the general solution is