© christine crisp “teach a level maths” vol. 2: a2 core modules 47: solving differential...

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules

47: Solving Differential 47: Solving Differential EquationsEquations

Solving Differential Equations

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Module C4


A differential equation is an equation which

contains a derivative such as or . dx

dy

dt

dA

e.g. (1) xdx

dy e.g. (2) y

dx

dy

Solving a differential equation means finding an expression for y in terms of x or for A in terms of t without the derivative.

To solve (1) we just integrate with respect to x.

Cx

yxdx

dy

2

2

However, we can’t integrate y w.r.t. x so (2) needs another method.


Before we see how to solve the equation, it’s useful to get some idea of the solution.

ydx

dye.g. (2)

The equation tells us that the graph of y has a gradient that always equals y.

We can sketch the graph by drawing a

gradient diagram. For example, at every point where y = 2, the gradient equals 2. We can draw a set of small lines showing this gradient.

2 1

We can cover the page with similar lines.


ydx

dy

We can now draw a curve through any point following the gradients.


ydx

dy

However, we haven’t got just one curve.


ydx

dy

The solution is a family of curves.

Can you guess what sort of equation these curves represent ?

ANS: They are exponential curves.


ydxdy Solvin

gWe use a method called “ Separating the Variables” and the title describes exactly what we do.

ydx

dydxdy

y

1

We rearrange so that x terms are on the right and y on the left.

Now insert integration signs . . .

dxdyy

1and integrate

Cxy ln

We can separate the 2 parts of the derivative because although it isn’t actually a fraction, it

behaves like one.

(the l.h.s. is integrated w.r.t. y and the r.h.s. w.r.t. x)

Multiply by dx and divide by y.

We don’t need a constant on both sides as they can be combined. I usually put it on the r.h.s.


ydx

dy

We’ve now solved the differential equation to find the general solution but we have an implicit equation and we often want it to be explicit ( in the form y = . . . )

Cxy ln

A log is just an index, so

Cxyln Cxey

( We now have the exponential that we spotted from the gradient diagram. )

However, it can be simplified.


So, xkeyydx

dy

We can write as . Cxe Cx ee

where k is positive

This is usually written as where A is positive or negative.

xAey

Cxey

So, ydx

dy xAey In this type of example, because the result is valid for positive and negative values, I usually use A directly when I change from log to exponential form.

Since is a constant it can be replaced by a single letter, k.

Ce


Changing the value of A gives the different curves we saw on the gradient diagram.

xAeyydx

dy

e.g. A = 2 gives


ydx

dyThe differential equation is important as

it is one of a group used to model actual

situations. These are situations where there is exponential growth or decay.

We will investigate them further in the next presentation.

We will now solve some other equations using the method of separating the variables.


xydx

dycos2

e.g. 3 Solve the equation below giving the answer in the form .)(xfy

Solution:

Separating the variables:

dxxdyy

cos1

2

Insert integration signs:

dxxdyy cos2

Cxy

sin1

1

Integrate:

Cxy

sin1

Cxy

sin

1


xyxdx

dy


Solution:

It’s no good dividing by y as this would give

xy

xdy

y

1 which is no help.

Instead, we take out x as a common factor on

the r.h.s., so )1( yxdx

dy

We can now separate the variables by dividing by

)1( y

xyxdx

dy


)1( yxdx

dy

Cx

y 2

1ln2

dxxdyy

1

1

Ay 1 2

2x

e

Cx

ey

2

2

1


)1( yxdx

dy

Cx

y 2

1ln2

dxxdyy

1

1

You may sometimes see this written as

2

exp2x

Ay 1 2

2x

e

Cx

ey

2

2

1


)1( yxdx

dy

Cx

y 2

1ln2

12

2

x

Aey

dxxdyy

1

1

Ay 1 2

2x

e

Cx

ey

2

2

1


dxxgdyyf )()(

SUMMARY• Some differential equations can be solved

by separating the variables.

• To use the method we need to be able to write the equation in the form

( If the equation has a total of 3 terms we will need to bracket 2 together before separating the variables. )• The l.h.s. is integrated w.r.t. y and the r.h.s.

w.r.t. x, so

dxxgdyyf )()(• The answer is often written

explicitly.• The solution is called the general solution.


Exercise

(a) xdx

dyx 2)1( 2 (b) 1

dx

dye y

2. Find the general solutions of the following equations leaving the answers in implicit form:(a) x

dx

dyy cossin b)

3. Find the equation of the curve given by the following equation and which passes through the given point.

0,0:2 2 yxexdx

dy y

1. Find the general solutions of the following equations giving your answers in the form :

)(xfy

( This is called a particular solution. )

yyxdx

dy 2


xdx

dyx 2)1( 2

Solutions:

dxx

xdy

21

2

Cxy )1ln( 2

(b) 1dx

dye y

dxdye y

Cxe y Cxy ln

dxxf

xf

)(

)(


2(a)

xdx

dyy cossin

dxxdyy cossin Cxy sincos

b) yyxdx

dy 2

dxxdyy

)1(1 2

Cxx

y 3

ln3

)1( 2 xydx

dy


0,0:2 2 yxexdx

dy y3.

dxxdye y

221

dxxdye y 22 C

xe y

3

2 3

0,0 yx C1

13

2 3

x

e yor 1

3

2ln

3

x

y

You might prefer to write as before you separate the variables.

ye ye

1


The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.


ydxdy e.g. 1 Solving

We use a method called “ Separating the Variables” and the title describes exactly what we do.

ydx

dydxdy

y

1

We rearrange so that x terms are on the right and y on the left.

Now insert integration signs . . .

dxdyy

1and integrate

Cxy ln

(the l.h.s. is integrated w.r.t. y and the r.h.s. w.r.t. x)

Multiply by dx and divide by y.


ydx

dy

We’ve now solved the differential equation to find the general solution but we have an implicit equation and we often want it to be explicit ( in the form y = . . . )

Cxy ln

A log is just an index, so

Cxyln Cxey

We usually use A.

So, xAeyydx

dy

We can write as . Since is a constant it can be replaced by a single letter.

Cxe Cx ee Ce

This can be simplified.


xydx

dycos


Solution:

Separating the variables:

dxxdyy

cos1

Insert integration

signs:

dxxdyy cos1

Cxy sinlnIntegrate:Cxey sinxAey sin


xyxdx

dy


Solution:

It’s no good dividing by y as this would give

xy

xdy

y

1 which is no help.

Instead, we take out x as a common factor on

the r.h.s. )1( yxdx

dy

We can now separate the variables by dividing by

)1( y


)1( yxdx

dy

Cx

y 2

)1ln(2

Cx

ey

2

2

1

2

2

1x

Aey

12

2

x

Aey

dxxdyy

1

1


dxxgdyyf )()(

SUMMARY• Some differential equations can be solved

by separating the variables.

• To use the method we need to be able to write the equation in the form

( If the equation has a total of 3 terms we will need to bracket 2 together before separating the variables. )• The l.h.s. is integrated w.r.t. y and the r.h.s.

w.r.t. x, so

dxxgdyyf )()(• The answer is often written

explicitly.• The solution is called the general solution.

© christine crisp “teach a level maths” vol. 2: a2 core modules 47: solving differential...

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