Identifying a second-order ODE

Undoubtedly0

Is there a general solution to

[tex] \frac{d}{dt}\left[p(t)\frac{dx(t)}{dt}\right] + q(t)x(t) = 0 [/tex]

for [itex] x(t) [/itex] when [itex] p(t) [/itex] and [itex] q(t) [/itex] are arbitrary functions? Better yet, does this question have a name, or some identifier, that I could look in to? It might appear more familiar written as

[tex] \left[p(t)x^\prime\right]^\prime + q(t)x = 0 [/tex]

JJacquelin

Quote by Undoubtedly0

Is there a general solution to
[tex] \frac{d}{dt}\left[p(t)\frac{dx(t)}{dt}\right] + q(t)x(t) = 0 [/tex]
for [itex] x(t) [/itex] when [itex] p(t) [/itex] and [itex] q(t) [/itex] are arbitrary functions?

No, there is no genertal solution
.

Chestermiller

Quote by Undoubtedly0

Is there a general solution to

[tex] \frac{d}{dt}\left[p(t)\frac{dx(t)}{dt}\right] + q(t)x(t) = 0 [/tex]

for [itex] x(t) [/itex] when [itex] p(t) [/itex] and [itex] q(t) [/itex] are arbitrary functions? Better yet, does this question have a name, or some identifier, that I could look in to? It might appear more familiar written as

[tex] \left[p(t)x^\prime\right]^\prime + q(t)x = 0 [/tex]

Look up Sturm-Liouville problems or equations.

GarageDweller

I think that's just a ODE with non constant coeffecients, since expanding yields

P(t)x''+P'(t)x'+q(t)x=0

You may be able to solve this with power series if P and q fit them.
Non linear differential equations rarely have closed form solutions.
But that's okay, we have computers

JJacquelin

Of course, when I say "There is no general solution", I mean "No general analytical solution espressed on a closed form".
Obviously, in some particular cases, with some particular forms of functions p(t) and q(t), the solutions might be known on closed form, and/or be expressed as infinite series.
Even more generally the solutions can be accurately approached thanks to numerical methods.

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GarageDweller	Identifying a second-order ODE I think that's just a ODE with non constant coeffecients, since expanding yields P(t)x''+P'(t)x'+q(t)x=0 You may be able to solve this with power series if P and q fit them. Non linear differential equations rarely have closed form solutions. But that's okay, we have computers

JJacquelin	Of course, when I say "There is no general solution", I mean "No general analytical solution espressed on a closed form". Obviously, in some particular cases, with some particular forms of functions p(t) and q(t), the solutions might be known on closed form, and/or be expressed as infinite series. Even more generally the solutions can be accurately approached thanks to numerical methods.