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second order ordinary differential equation
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(Topic)
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A second order ordinary differential equation $F(x,\,y,\,\frac{dy}{dx},\,\frac{d^2y}{dx^2}) = 0$ can often be written in the form
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(1) |
By its general solution one means a function $x \mapsto y = y(x)$ which is at least on an interval twice differentiable and satisfies $$ y''(x) \;\equiv\; f(x,\,y(x),\,y'(x)) $$ By setting $\frac{dy}{dx} := z$ , one has $\frac{d^2y}{dx^2} = \frac{dz}{dx}$ , and the equation (1) reads $\frac{dz}{dx} = f(x,\,y,\,z)$ . It is easy to see that solving (1) is equivalent with solving the system of simultaneous first order differential equations
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(2) |
the so-called normal system of (1).
The system (2) is a special case of the general normal system of second order, which has the form
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(3) |
where $y$ and $z$ are unknown functions of the variable $x$ . The existence theorem of the solution
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(4) |
is as follows; cf. the Picard-Lindelöf theorem.
Theorem. If the functions $\varphi$ and $\psi$ are continuous and have continuous partial derivatives with respect to $y$ and $z$ in a neighbourhood of a point $(x_0,\,y_0,\,z_0)$ , then the normal system (3) has one and (as long as $|x\!-\!x_0|$ does not exceed a certain bound) only one solution (4) which satisfies the
initial conditions $y(x_0) = y_0,\;\, z(x_0) = z_0$ . The functions (4) are continuously differentiable in a neighbourhood of $x_0$ .
- 1
- E. LINDELÖF: Differentiali- ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
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"second order ordinary differential equation" is owned by pahio.
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normal system, normal system of second order |
This object's parent.
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Cross-references: continuously differentiable, initial conditions, point, neighbourhood, partial derivatives, continuous, theorem, solution, existence theorem, variable, differential equations, easy to see, equation, twice differentiable, interval, function, general solution
There are 4 references to this entry.
This is version 2 of second order ordinary differential equation, born on 2008-12-07, modified 2008-12-08.
Object id is 11322, canonical name is SecondOrderOrdinaryDifferentialEquation.
Accessed 958 times total.
Classification:
| AMS MSC: | 34A05 (Ordinary differential equations :: General theory :: Explicit solutions and reductions) |
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Pending Errata and Addenda
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