MATH 308 Lecture 1

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Lecture Notes

Differential Equations

An equation involving derivatives

Vocabulary

mathematical model

Equations that describes some physical phenomena

order

the highest derivative taken in the equation

Notation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}^n y}{\mathrm{d} x^n} = y^{(n)}}

Examples

Motion of a Spring

${\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-kx(t)$

ordinary differential equation (ODE) of order 2

$t$ independent

$x$ dependent

Pendulum

${\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=-\sin {\theta }$

ODE of order 2

this equation is non-linear because of $\sin {\theta }$ (therefore not easy to solve as-is)

approx. for small $\theta$ ($\sin {\theta }\approx \theta$) is linear

$t$ independent

$\theta$ dependent

Vibrating string (propagation of waves)

${\frac {\partial ^{2}u}{\partial x^{2}}}-c^{2}\,{\frac {\partial ^{2}u}{\partial t^{2}}}=0$

partial differential equation (PDE) of order 2

$x,t$ independent

$u$ dependent

Slightly Complicated

$y^{(5)}(x)+x^{2}\,y^{(3)}(x)-5\mathrm {e} ^{x}\,y'=\sin {x}$

ODE of order 5

$x$ independent

$y$ dependent

Simple

${\frac {\mathrm {d} f}{\mathrm {d} x}}=f(x)$

ODE of order 1

$x$ is independent variable

$f$ is dependent variable

Linearity

If you can write an ODE in the format

$\sum _{i=0}^{n}a_{i}(x)y^{(i)}(x)=\sum _{i=0}^{n}a_{i}(x){\frac {\mathrm {d} ^{i}y}{\mathrm {d} x^{i}}}=g(x)$

$g(x)$ and each $a_{i}(x)$ do not (and cannot) depend on $y$

Linear differential equations are easier to solve than non-linear diff. eq's.

Exercises

Classify each as linear or non-linear

1. $y''+y\,\sin {x}=x^{3}$ linear
2. $y''+x\,\sin {y}=x^{2}$ non-linear
3. $y''+x^{2}\,y={\sqrt {x}}$ linear
4. $y''+x^{2}\,y'={\sqrt {y}}$ non-linear
5. $y''+y\,y'={\sqrt {x}}$ non-linear

Solutions

A function $f$ is a solution to a differential equation if $f$ satisfies the differential equation

Show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = 3 \sin{\left( 2x \right)} + \mathrm{e}^{-x}} is a solution to the differential equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'' + 4y = 5e^{-x}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} y'' &= -12 \sin{\left(2x\right)} + \mathrm{e}^{-x} \\ \left(-12 \sin{\left(2x\right)} + \mathrm{e}^{-x}\right) + 4 \left(3 \sin{\left(2x\right)} + \mathrm{e}^{-x} \right) &= 5\mathrm{e}^{-x} \end{align}}

Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = 3 \sin{\left( 2x \right)} + \mathrm{e}^{-x}} is a solution to the differential equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'' + 4y = 5e^{-x}} .