Relationship between DEs and infinite series.

[ellipsis]

Not all DEs have a closed form solution. Some DEs have an implicit solution only - you cannot algebraically solve one variable of interest for another.

I have seen on this forum people solving DEs in terms of infinite series. How does one arrive at such a solution, and can an implicit solution be converted somehow to an infinite series solution? Is it possible to solve any DE in terms of a single or multiple infinite series?

 

[DeIdeal]

For a practical example, one can solve second-order ODEs (with sufficiently nice properties) by using the Frobenius method. There's no general way of constructing a series solution to an arbitrary DE, the procedure depends on the type of the equation.

There are obviously DEs that have no solutions at all, so the answer to your last question would be a somewhat trivial no.

 

[ellipsis]

Know of any method for the following?
$$
\frac{d^2x}{dt^2} = 1 - \frac{1}{(1+x)^2} - (\frac{dx}{dt})^2
$$

 

[DeIdeal]

By hand? Not in general, but in that specific case the solutions seem to be logarithms of modified Bessel functions, so I'd instinctively let y(x(t))=exp(x(t)) and try to solve the resulting equation with the Frobenius method (or just try to get the modified Bessel equation out). Haven't tried it through, so take this with a grain of salt.