How can I understand the result Mathematica returns from DSolve?

Question

I have solved an ODE using DSolve, but I have a problem with understanding the solution. In general the solution is in the form:

InverseFunction[[many expressions using # and #1]&][g x+C[1]]

where g is constant

What does #1 and & mean and what does InverseFunction mean in this context?

Answer 1

Some DEs are more simple to solve for the dependent variable rather than the independent variable, for example $$ \frac{dy}{dx} = y \quad\implies\quad \log(y)=x+c $$ from which you can obtain the solution for $y$ in terms of $x$ by using an inverse function, in this case $y=\exp(x+c)$. Not all examples are this easy to invert, so Mathematica sometimes has to leave the solution written in terms of InverseFunction.

The # and & are part of Mathematica's pure (or anonymous) function notation. In particular & occurs at the end of a pure function and #=#1 represents the first slot of the function. For example

(#^2 + 1&)

is equivalent to

Function[{x}, x^2 + 1]

and acts upon its arguments like any other function

(#^2 + 1&)[t] == Function[{x}, x^2 + 1][t] == t^2 + 1

So, your DE must have yielded a complicated algebraic expression $f(y)=x+c$ that needs to be solved for the variable that you are interested in, $y=f^{(-1)}(x+c)$, which Mathematica can only perform symbolically using InverseFunction.