# Solution of Equations

# Polynomic Equations

## Quadratics

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## Cubics

_{}

Let

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_{}

_{}

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The three solutions are

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_{}

# Differential Equations

## Separation of Variables

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## Exact Equation

_{} _{}

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## Linear First Order Equation

_{}

_{}

## Bernoulli’s Equation

_{}

Let

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Apply the recursive formula,

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If _{}, the solution is

_{}

## Homogeneous Equation

_{}

Let

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For _{}, the solution is

_{}

If _{}, the solution is

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## Linear, Homogeneous, Second Order Equation

_{}

*b* and *c* are real constants

Find the roots, _{} and _{}, of the equation,

_{}

There are three cases:

Case 1: _{} and _{} are real and distinct

_{}

Case 2: _{} and _{} are real and equal

_{}

Case 3: _{} and _{} are imaginary

Let

_{}

_{}

The solution is

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## Linear, Nonhomogeneous, Second Order Equation

_{}

*b* and *c* are real constants

Find the roots, _{} and _{}, of the equation,

_{}

There are three cases:

Case 1: _{} and _{} are real and distinct

_{}

Case 2: _{} and _{} are real and equal

_{}

Case 3: _{} and _{} are imaginary

Let

_{}

_{}

The solution is

_{}

## Euler or Cauchy Equation

_{}

Let

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The equation becomes a linear second-order equation

_{}

## Bessel’s Equation

_{}

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## Transformed Bessel’s Equation

_{}

Let

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## Legendre’s Equation

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