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Consider a second order homogeneous ODE: $$P(x)y''+Q(x)y'+R(x)y=0.$$ If $P,Q,R$ are constant functions, then we know that the general solution has the form $$y=c_1e^{r_1x}+c_2e^{r_2x},$$ $$y=c_1e^{r_1x}+c_2xe^{r_2x},$$ or $$y=e^{\alpha x}[c_1\cos(\beta x)+c_2\sin(\beta x)],$$ depending on the constants $P,Q,R$.


If $P,Q,R$ are not constant, the solutions are not so simple. However, most resources I could find (such as this one) do not discuss this case at all. What I was wondering is, even if we can't solve the equation directly, is it possible to get some meaningful information on the behaviour of the solutions? For example, is it possible to tell that the solution "behaves like an exponential" asymptotically, or oscillates like a sine/cosine function? Any reference discussing such questions would be welcome.

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    $\begingroup$ please check your typesetting. $\endgroup$ – MrYouMath May 24 '16 at 14:31
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    $\begingroup$ Are you looking for the theory of linear dynamical systems? $\endgroup$ – Mattos May 24 '16 at 14:35
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    $\begingroup$ I don't think you're going to find a nice universal answer here. If you allow for arbitrary $P,Q,R$ then this allows for singular points, regular and otherwise, and the general behavior allowed has many facets. You get series of algebraic functions, logs of trig functions, Bessel functions and a host of other functions that need not (or simply are not) named. $\endgroup$ – James S. Cook May 24 '16 at 14:41
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    $\begingroup$ @RaisinBread A good place to start is "Differential Dynamical Systems" by J. D. Meiss. $\endgroup$ – Mattos May 24 '16 at 14:42
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    $\begingroup$ Up until today there is no general solution to this linear 2nd order ODE with non-constant coefficients. You can use perturbation theory to find approximate or asymptotic solutions. $\endgroup$ – MrYouMath May 24 '16 at 14:42
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Let $f$ be a twice differentiable function. Observe $$ \text{det} \left[ \begin{array}{ccc} y & y' & y'' \\ f(x) & f'(x) & f''(x) \\ e^x & e^x & e^x \end{array} \right] =0$$ is a 2nd order linear ODE with horrible coefficients which takes $y=f(x)$ as a solution (and $y=e^x$ as a second solution). But, $f(x)$ is nearly arbitrary, so your questions is really a question about characterizing essentially arbitrary functions in terms of exponentials etc.

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  • $\begingroup$ Of course, you can replace $e^x$ with a different function and its derivatives if you like, the identity stems from the fact that a repeated row in a determinant produces a result of zero. $\endgroup$ – James S. Cook May 24 '16 at 14:50

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