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?


1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.