Non linear to linear differential equation

Question

Is there exist a non linear differential equation $$ y'(t) = f(t,y(t)) $$ such that a change of variable $z=\varphi(t,y)$ leads to a linear differential equation ?

Answer 1

Yes, of course. You can obtain such a nonlinear equation if you apply a nonlinear change of variable to a linear equation.

Consider, for example, the equation $$ \dot z =z,\quad z\in\mathbb R $$ and the transformation $$ y=\arctan z,\quad y\in \left(-\frac{\pi}2;\frac{\pi}2\right). $$ Using the chain rule we can obtain $$ \dot y= \frac{\dot z}{1+z^2}=\frac{z}{1+z^2}=\tan y\cos^2 y=\sin y\cos y $$ The resulting nonlinear equation $$ \dot y=\sin y\cos y,\quad y\in \left(-\frac{\pi}2;\frac{\pi}2\right) $$ can be transformed back to the linear form using the change of variable $z= \tan y$