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: $\frac{{\mathrm{d}}^{n}y}{\mathrm{d}{x}^n}=y^{(n)}$

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

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

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. $y^{″}+x^2{y}^{\prime }=\sqrt{y}$ non-linear
5. $y^{″}+y{y}^{\prime }=\sqrt{x}$ non-linear

Solutions

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

Show that $y=3\sin \left(2x\right)+{\mathrm{e}}^{-x}$ is a solution to the differential equation $y^{″}+4y=5e^{-x}$

$\begin{array}{rl}{y}^{″}& =-12\sin \left(2x\right)+{\mathrm{e}}^{-x}\\ (-12\sin \left(2x\right)+{\mathrm{e}}^{-x})+4(3\sin \left(2x\right)+{\mathrm{e}}^{-x})& =5{\mathrm{e}}^{-x}\end{array}$

Therefore, $y=3\sin \left(2x\right)+{\mathrm{e}}^{-x}$ is a solution to the differential equation $y^{″}+4y=5e^{-x}$.