47 Solving DifferentiDifferential Equation Equations

7/27/2019 47 Solving DifferentiDifferential Equation Equations

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© Christine Crisp

“Teach A Level Maths”

Vol. 2: A2 Core Modules

47: Solving Differential

Equations



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) x dx

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 .

C x y x dx 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 usefulto get some idea of the solution.

y dx

dy e.g. (2)

The equation tells us that the graph of y has a

gradient that always equalsy .We can sketch the graph by drawing a

gradient diagram.For example, at every

point wherey = 2

, thegradient equals 2. We candraw a set of small linesshowing this gradient.

2 1

We can cover the page with similar lines.




y

dx

dy

We can now draw a curve through any point followingthe gradients.




y

dx

dy

However, we haven’t got just one curve.




y

dx

dy

The solution is a family of curves.Can you guess what sort of equation these curves

represent ?

ANS: They are exponential curves.




y dx

dy Solving

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

y dx

dy dx dy

y

1

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

Now insert integration signs . . .

dx dy y

1

and integrate

C x y ln

We can separate the 2 parts of the derivative because

although it isn’t actually a fraction, it behaves likeone.

(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 theycan be combined. I usually put it on the r.h.s.




y dx

dy

We’ve now solved the differential equation to find thegeneral solution but we have an implicit equation and we

often want it to be explicit ( in the form y = . . . )

C x y ln

A log is just an index, so

C x y ln C x e y

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

However, it can be simplified.




So,

x

ke y y dx

dy

We can write as .C x e C x e e

where k is positiveThis is usually written as where A ispositive or negative.

x Ae y

C x e y

So, y dx

dy x

Ae y In this type of example, because the result is validfor 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 .

C e




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

x Ae y y

dx

dy

e.g. A = 2 gives




y

dx

dy The differential equation is important as it is

one of a group used to model actual situations.

These are situations where there is exponentialgrowth or decay.

We will investigate them further in the nextpresentation.

We will now solve some other equations using the

method of separating the variables.




x y dx

dy cos

2

e.g. 3 Solve the equation below giving the answer in

the form .)(x f y

Solution: Separating the variables: dx x dy y

cos12

Insert integration signs:dx x dy y cos

2

C x y

sin1

1

Integrate:

C x y

sin1

C x

y

sin

1




xy x dx

dy


the form.)(x f y

Solution:

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

x y

x dy

y

1which is no help.

Instead, we take out x as a common factor on the

r.h.s., so)1( y x

dx

dy

We can now separate the variables by dividing by )1( y

xy x dx

dy




)1( y x dx

dy

C

x

y 21ln

2

dx x dy y

1

1

Ay 1 2

2x

e

C x

e y

2

2

1




)1( y x dx

dy

C

x

y 21ln

2

dx x dy y

1

1

You may sometimes

see this written as

2exp

2x

Ay 1 2

2x

e

C x

e y

2

2

1




)1( y x dx

dy

C

x

y 21ln

2

12

2

x

Ae y

dx x dy y

1

1

Ay 1 2

2x

e

C x

e y

2

2

1




2(a) x dx

dy y cossin

dx x dy y cossin C x y sincos

b) y yx dx

dy 2

dx x dy y

)1(1

2 C x

x y

3ln

3

)1(2 x y

dx

dy




0,0:22

y x e x dx

dy y 3.

dx x dy e

y

221

dx x dy e y 22

C x

e y

3

2 3

0,0 y x C 1

13

23

x e y or 13

2ln3

x y

You might prefer to write as before you

separate the variables.

y e

y e

1




The following slides contain repeats ofinformation on earlier slides, shown withoutcolour, so that they can be printed and

photocopied.

For most purposes the slides can be printedas “Handouts” with up to 6 slides per sheet.




x y dx

dy cos


the form .)(x f y

Solution: Separating the variables: dx x dy

y

cos1

Insert integration signs:

dx x dy y cos1

C x y sinlnIntegrate:C x

e y sin

x Ae y

sin




xy x dx

dy


the form .)(x f y

Solution:

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

x y

x dy

y

1which is no help.

Instead, we take out x as a common factor on the

r.h.s.)1( y x

dx

dy

We can now separate the variables by dividing by )1( y




)1( y x dx

dy

C

x

y 2)1ln(

2

C x

e y

2

2

1

2

2

1x

Ae y

12

2

x

Ae y

dx x dy y

1

1




dx x g dy y f )()(

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 tobracket 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

dx x g dy y f )()(

• The answer is often written explicitly.

• The solution is called the general solution.

47 Solving DifferentiDifferential Equation Equations

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