First and Second Order Differential Equations

First Order Differential equations

A first order differential equation is of the form:

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Linear Equations:

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The general general solution is given by

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where

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is called the integrating factor.

Separable Equations:

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(1)
Solve the equation g(y) = 0 which gives the constant solutions.
(2)
The non-constant solutions are given by

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Bernoulli Equations:

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(1)
Consider the new function tex2html_wrap_inline153 .
(2)
The new equation satisfied by v is

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(3)
Solve the new linear equation to find v.
(4)
Back to the old function y through the substitution tex2html_wrap_inline163 .
(5)
If n > 1, add the solution y=0 to the ones you got in (4).

Homogenous Equations:

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is homogeneous if the function f(x,y) is homogeneous, that is

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By substitution, we consider the new function

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The new differential equation satisfied by z is

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which is a separable equation. The solutions are the constant ones f(1,z) - z =0 and the non-constant ones given by

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Do not forget to go back to the old function y = xz.

Exact Equations:

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is exact if

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The condition of exactness insures the existence of a function F(x,y) such that

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All the solutions are given by the implicit equation

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Second Order Differential equations

Homogeneous Linear Equations with constant coefficients:

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Write down the characteristic equation

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(1)
If tex2html_wrap_inline201 and tex2html_wrap_inline203 are distinct real numbers (this happens if tex2html_wrap_inline205 ), then the general solution is

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(2)
If tex2html_wrap_inline209 (which happens if tex2html_wrap_inline211 ), then the general solution is

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(3)
If tex2html_wrap_inline201 and tex2html_wrap_inline203 are complex numbers (which happens if tex2html_wrap_inline219 ), then the general solution is

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where

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that is

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Non Homogeneous Linear Equations:

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The general solution is given by

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where tex2html_wrap_inline231 is a particular solution and tex2html_wrap_inline233 is the general solution of the associated homogeneous equation

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In order to find tex2html_wrap_inline237 two major techniques were developed.

Method of undetermined coefficients or Guessing Method

This method works for the equation

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where a, b, and c are constant and

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where tex2html_wrap_inline249 is a polynomial function with degree n. In this case, we have

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where

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The constants tex2html_wrap_inline257 and tex2html_wrap_inline259 have to be determined. The power s is equal to 0 if tex2html_wrap_inline265 is not a root of the characteristic equation. If tex2html_wrap_inline265 is a simple root, then s=1 and s=2 if it is a double root.
Remark. If the nonhomogeneous term g(x) satisfies the following

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where tex2html_wrap_inline277 are of the forms cited above, then we split the original equation into N equations

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then find a particular solution tex2html_wrap_inline283 . A particular solution to the original equation is given by

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Method of Variation of Parameters

This method works as long as we know two linearly independent solutions tex2html_wrap_inline287 of the homogeneous equation

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Note that this method works regardless if the coefficients are constant or not. a particular solution as

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where tex2html_wrap_inline293 and tex2html_wrap_inline295 are functions. From this, the method got its name.
The functions tex2html_wrap_inline293 and tex2html_wrap_inline295 are solutions to the system:

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which implies

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Therefore, we have

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Euler-Cauchy Equations:

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where b and c are constant numbers. By substitution, set

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then the new equation satisfied by y(t) is

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which is a second order differential equation with constant coefficients.

(1)
Write down the characteristic equation

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(2)
If the roots tex2html_wrap_inline201 and tex2html_wrap_inline203 are distinct real numbers, then the general solution is given by

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(2)
If the roots tex2html_wrap_inline201 and tex2html_wrap_inline203 are equal ( tex2html_wrap_inline209 ), then the general solution is

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(3)
If the roots tex2html_wrap_inline201 and tex2html_wrap_inline203 are complex numbers, then the general solution is

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where tex2html_wrap_inline339 and tex2html_wrap_inline341 .

[Differential Equations]
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