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

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

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

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

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