001 Getting started with some definitions and examples.ipynb

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   "source": [
    "# Getting started"
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   "source": [
    "## In this lesson"
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    "- [A differential equation](#A-differential-equation)\n",
    "- [The order of a differential equation](#The-order-of-a-differential-equation)\n",
    "- [The linearity of a differential equation](#The-linearity-of-a-differential-equation)"
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   "source": [
    "## A differential equation"
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    "A differential equation contains the derivative of one or more of the dependent variables with respect to one or more of the independent variables."
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    "Examples of differntial equations:\n",
    "$$ y' \\left( t \\right) - y \\left( t \\right) = 0 \\\\ \\frac { { \\partial  }^{ 2 } }{ \\partial { t }^{ 2 } } f\\left( t \\right) +2\\frac { \\partial  }{ \\partial t } f\\left( t \\right) +4=0\\\\ \\frac { { d }^{ 2 }y }{ { dt }^{ 2 } } -\\frac { dy }{ dt } +2y=0\\\\ \\frac { dy }{ dt } +\\frac { dx }{ dt } -2y-2x=0 $$"
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   "source": [
    "Note the different notations and the fact that we can have higher-order differantials and more than one dependent variable."
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    "All the example above are actually **ordinary differential equations** (ODEs).  We also get **partial differential equations** (PDEs).  These involve partial derivatives of one or more dependent variable with respect to two or more independent variables.  Here are some examples:\n",
    "$$ \\frac { { \\partial  }^{ 2 }u }{ \\partial { x }^{ 2 } } +\\frac { { \\partial  }^{ 2 }u }{ \\partial { y }^{ 2 } } =0\\\\ \\frac { \\partial u }{ \\partial y } +\\frac { \\partial v }{ \\partial x } =0 $$"
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    "[Back to the top](#In-this-lesson)"
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    "## The order of a differential equation"
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    "The order of a differntial equation is given by the highest derivative in the equation.  Written in prime notation the equation $ f'''\\left( t \\right) + 9 f'' \\left( t \\right) = 0 $ would have order $ 3 $, also then called a third-order ODE."
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    "This helps us formulate a common notation for differential equations.  Look at these:\n",
    "$$ \\frac { dy }{ dt } =f\\left( t,y \\right) \\\\ \\frac { { d }^{ 2 }y }{ { dt }^{ 2 } } =f\\left( t,y,y' \\right) \\\\ \\frac { { d }^{ 3 }y }{ { dt }^{ 3 } } =f\\left( t,y,y',y'' \\right) \\\\ \\vdots \\\\ \\frac { { d }^{ n }y }{ { dt }^{ n } } =f\\left( t,y,y',y'',\\dots ,{ y }^{ \\left( n-1 \\right)  } \\right)   $$"
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    "This is called the **normal form** allowing us to write $ y'' \\left( t \\right) - 3 y \\left( t \\right) + 2 = 0 $ as $ y'' \\left( t \\right) = 3 y \\left( t \\right) - 2 $."
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   "source": [
    "[Back to the top](#In-this-lesson)"
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    "## The linearity of a differential equation"
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    "An $ n $<sup>th</sup> order ODE is linear if all the expressions of the dependent variable and all the derivatives of the dependent variable are linear, i.e. to the power one.  The following are therefore not linear ODEs:\n",
    "$$ \\frac { { d }^{ 2 }y }{ { dt }^{ 2 } } +\\frac { dy }{ dt } +{ y }^{ 2 }=0\\\\ \\frac { { d }^{ 2 }y }{ { dt }^{ 2 } } +\\frac { dy }{ dt } +{ e }^{ y }=0 $$"
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    "[Back to the top](#In-this-lesson)"
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