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.

 

[blue_raver22]

The relationship between differential equations (DEs) and infinite series is a fascinating topic in mathematics. While not all DEs have a closed form solution, some can be solved using infinite series. This approach is known as power series or Taylor series method.

To arrive at an infinite series solution, one must first rewrite the DE as a power series. This involves expressing the dependent variable and its derivatives as a sum of powers of the independent variable. This series is then substituted into the DE, and by equating the coefficients of each power, a system of equations is obtained. Solving this system will give the coefficients of the infinite series, which can then be used to approximate the solution of the DE.

However, it is important to note that not all DEs can be solved using this method. The convergence of the infinite series solution depends on the behavior of the DE and the initial conditions. In some cases, the series may not converge, or it may converge to a different solution than the one obtained by other methods.

As for converting an implicit solution to an infinite series solution, it is possible in some cases. This involves manipulating the implicit solution to express it in terms of a power series. However, this process can be challenging and may not always be successful.

In conclusion, while infinite series solutions offer an alternative approach to solving DEs, it is not always possible to solve any DE in terms of a single or multiple infinite series. The convergence of the series and the ability to convert an implicit solution to an infinite series solution depend on the specific DE and its initial conditions.