# 1. Solving Differential Equations (DEs)

**differential equation**(or "DE") contains

**derivatives**or

**differentials**.

**solve**the differential equation. This will involve integration at some point, and we'll (mostly) end up with an expression along the lines of "

*y*= ...".

**derivative**where $\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}$ is actually not written in fraction form.

### Examples of Differentials

###
Examples of Differential Equations*dx* (this means "an infinitely small change in *x*")

$\displaystyle{d}\theta$ (this means "an infinitely small change in $\displaystyle\theta$")

$\displaystyle{\left.{d}{t}\right.}$ (this means "an infinitely small change int")

### Example 1

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={x}^{2}-{3}$x2−3

**general solution**(involving

*K*, a constant of integration).

$\displaystyle{y}=\int{\left({x}^{2}-{3}\right)}{\left.{d}{x}\right.}$

$\displaystyle{y}=\frac{{x}^{3}}{{3}}-{3}{x}+{K}$

*dy*go from the $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$? Why did it seem to disappear?

*x*part only (on the right), but in fact we have integrated with respect to

*y*as well (on the left). DEs are like that - you need to integrate with respect to two (sometimes more) different variables, one at a time.

**differentials**:

dy= (x^{2}− 3)dx

*dy/dx*in the question by

*dx*.)

*y*(that's why we use "

*dy*") and the right side with respect to

*x*(that's why we use "

*dx*") :

$\displaystyle\int{\left.{d}{y}\right.}=\int{\left({x}^{2}-{3}\right)}{\left.{d}{x}\right.}$

*dy*part more carefully:

$\displaystyle{y}=\frac{{x}^{3}}{{3}}-{3}{x}+{K}$

*y.*

**Note about the constant:**We have integrated both sides, but there's a constant of integration on the right side only. What happened to the one on the left? The answer is quite straightforward. We do actually get a constant on both sides, but we can combine them into one constant (

*K*) which we write on the right hand side.

### Example 2

$\displaystyle\theta^{2}{d}\theta= \sin{{\left({t}+{0.2}\right)}}{\left.{d}{t}\right.}$

A function of $\displaystyle\theta$ with $\displaystyle{d}\theta$ on the left side, and

A function oftwithdton the right side.

$\displaystyle\int\theta^{2}{d}\theta=\int \sin{{\left({t}+{0.2}\right)}}{\left.{d}{t}\right.}$

$\displaystyle\frac{{\theta^{3}}}{{3}}=- \cos{{\left({t}+{0.2}\right)}}+{K}$K

*t*on the right.

## Solving a differential equation

**solving a DE**means finding an equation with no derivatives that satisfies the given DE. Solving a differential equation always involves one or more

**integration**steps.

**type of DE**we are dealing with before we attempt to solve it.

## Definitions

**First order DE:**Contains only first derivatives

**Second order DE:**Contains second derivatives (and possibly first derivatives also)

**Degree:**The

**highest power**of the

**highest derivative**which occurs in the DE.

### Example 3 - Order and Degree

**order 2**(the highest derivative appearing is the

**second**derivative) and

**degree 1**(the

**power**of the highest derivative is 1.)

**order 1**(the highest derivative appearing is the

**first**derivative) and

**degree 5**(the

**power**of the highest derivative is 5.)

**order 2**(the highest derivative appearing is the

**second**derivative) and

**degree 4**(the

**power**of the highest derivative is 4.)

## General and Particular Solutions

**general solution**(involving a constant,

*K*).

**particular solution**by substituting known values for

*x*and

*y*. These known conditions are called

**boundary conditions**(or

**initial conditions**).

### Example 4

$\displaystyle{\left.{d}{y}\right.}+{7}{x}{\left.{d}{x}\right.}={0}$

### Example 5

$\displaystyle{y}'={5}$

### Example 6

$\displaystyle{y}{'''}={0}$

$\displaystyle{y}{\left({0}\right)}={3},$ $\displaystyle{y}'{\left({1}\right)}={4},$ $\displaystyle{y}{''}{\left({2}\right)}={6}$

### Example 7

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}} \ln{{x}}-\frac{y}{{x}}={0}$xy=0

$\displaystyle{y}={c} \ln{{x}}$

## Second Order DEs

### Example 8

y'' +a^{2}y= 0

$\displaystyle{y}={A} \cos{{a}}{x}+{B} \sin{{a}}{x}$

### Example 9

y'' − 3y' + 2y= 0

y=Ae^{2x}+Be^{x}

y(0) = 4,y'(0) = 5

y=e^{2x}+ 3e^{x}

Now we do some examples using second order DEs where we are given a final answer and we need to check if it is the correct solution.

### Example 10 - Second Order DE

$\displaystyle{y}={c}_{{1}} \sin{{2}}{x}+{3} \cos{{2}}{x}$

$\displaystyle\frac{{{d}^{2}{y}}}{{{\left.{d}{x}\right.}^{2}}}+{4}{y}={0}$+4y=0

### Example 11 - Second Order DE

*y*=

*c*

_{1}+

*c*

_{2}

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