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equation 
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(Topic)
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A simple special case of the second order linear differential equation with constant coefficients is
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where $f$ is continuous. We obtain immediately $\displaystyle\frac{dy}{dx} = C_1+\!\int\!f(x)\,dx$ ,
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A particular solution $y(x)$ of (1) satisfying the initial conditions $$y(x_0) = y_0, \quad y'(x_0) = y_0'$$ is obtained more simply by integrating (1) twice between the limits $x_0$ and $x$ , thus getting $$y(x) = y_0+y_0'\!\cdot\!(x\!-\!x_0)+\!\int_{x_0}^x\!\left(\int_{x_0}^x f(x)\,dx\right)dx.$$
But here, the two first addends are the first terms of the Taylor polynomial of $y(x)$ , expanded by the powers of $x-x_0$ , whence the double integral is the corresponding remainder term $$\int_{x_0}^x y''(x)(x\!-\!t)\,dt \;=\; \int_{x_0}^x f(t)(x\!-\!t)\,dt.$$ Hence the particular solution can be written with
the simple integral as
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The result may be generalised for the $n^\mathrm{th}$ order differential equation
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with corresponding $n$ initial conditions:
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"equation  " is owned by pahio.
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Cross-references: differential equation, integral, double integral, powers, expanded, Taylor polynomial, terms, initial conditions, particular solution, continuous, second order linear differential equation with constant coefficients
There are 2 references to this entry.
This is version 7 of equation  , born on 2008-12-07, modified 2008-12-08.
Object id is 11320, canonical name is EquationYFx.
Accessed 492 times total.
Classification:
| AMS MSC: | 34-01 (Ordinary differential equations :: Instructional exposition ) | | | 34A30 (Ordinary differential equations :: General theory :: Linear equations and systems, general) |
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Pending Errata and Addenda
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