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The solution of a linear first-order ODE, $y'+P(x)y+Q(x)=0$, is expressible by integrals involving elementary functions, $P(x)$ and $Q(x)$. This can be proved e.g. by the applying the method of integrating factor to the equation.

This makes me wonder - are solutions to the general second-order ODE expreesible using integrals? We can concentrate only on homogenous equations - the inhomogenous case is accounted for by the method of variation of parameters. I could not find a similar formula in this case, so I suspect that none exists. However, even equations such as Bessel's equation admit solutions expressible by integrals, e.g. $J_n(x)=\frac 1 \pi \int_0^\pi \cos(nt-x\sin t) \,\text{d}t$. Is such a solution possible for every linear second-order ODE?

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  • $\begingroup$ In the case of $J_n$ the function is defined by an integral, but it is not an antiderivative, contrary to what the ingredients that appear in the case of first-order ODEs. $\endgroup$ – Loïc Teyssier Dec 3 '20 at 17:22
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    $\begingroup$ The theory that captures solvability by antiderivative is called "Differential Galois Theory" and indeed the generic second order linear equation is not solvable in that sense. More details are given in my previous answers mathoverflow.net/questions/181530/… and mathoverflow.net/questions/140849/solution-of-linear-ode/…. $\endgroup$ – Loïc Teyssier Dec 3 '20 at 17:22
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    $\begingroup$ But I don't know whether there is a theory capturing solvability by parametric integrals. $\endgroup$ – Loïc Teyssier Dec 3 '20 at 17:24
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    $\begingroup$ Already for $y''=a(x)y$ there's no method to find the solutions. $\endgroup$ – Christian Remling Dec 3 '20 at 21:03
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As a rule, the answer to your question is "no", in the sense that no closed form solution can be obtained, let alone one expressible by integrals.

Substituting $z = y^\prime /y$ into the linear second order equation $$y^{\prime\prime} + P(x)y^\prime+ Q(x)y =0$$ gives the Riccati equation $$z^\prime = -z^2 - P(x)z - Q(x)$$.

In general, this is not solvable. However, if the coefficients $P(x)$ and $Q(x)$ satisfy particular constraint conditions e.g. L. Bougoffa, New Conditions for Obtaining the Exact Solutions of the General Riccati Equation, (https://www.hindawi.com/journals/tswj/2014/401741/) a closed form solution can be obtained.

Per Loïc Teyssier's comment, a treatment of the Riccati equation using differential Galois theory has been carried out by A. Sebbar and O. Wone Anharmonic Solutions of the Riccati equation and modular functions https://arxiv.org/abs/1211.5661.

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