Implicit solutions to differential equations

Question

Ordinary differential equations are often of the form $f^{'}(x)=F(f(x),x)$. Solutions to such equations are functions that satisfy them and are differentiable.

If we integrate both sides of the equation we get $f(x)=f(0)+\int_{0}^{x}F(f(t),t)dt$, which looks like a solution but on the other hand we can also view it as an equivalent equation.

Which of those two things is it, is it both?

Answer 1

Anything called solution has to add some information. The transformation into the somewhat equivalent integral form is valid for all explicit order-1 ODE, as you carried out, thus no information gain.

Usually one understands under implicit solution the transformation into some implicit equation $0=G(x,f(x),C)$. This, while still no direct computation procedure for the value $f(x)$ from $x$ alone however provides more information as it allows to numerically compute or at least refine (from one Euler step from the last value for instance) the value $f(x)$ without reference to the whole history up to the initial point.

Answer 2

Both equations possibly define a function, but in an implicit way. A solution needs to be explicit.

$f'(x)=f(x)$ is an equation, $f(x)=e^x$ is a solution.