Differential Equations

Separation of variablesExact factors of Differential Equations

Integrating factors

Before you begin you should be confident with:Integration by parts, substitution and trigonometric functionsDifferential Equations by separation of variablesImplicit differentiation

ReminderSolve the following differential equations

1

111ln

111ln

11ln

11

1

1

11

2

2

xey

xy

cc

cx

y

dxx

dyy

ydxdyx

322

321

22

2

2

2

yyx

yyx

yx

yx

yx

eyee

cc

ceyee

dyyedxedxdyyee

dxdyyex=1, y=0 x=0, y=0

Integrate by parts

Try solving this differential equation

xxydxdyx sin322 When x=0 and

y=1

We cannot separate the variables in the above, however, you may have noticed that the LHS is the exact derivative of x2y..

xydxdyxyx

dxd 222

We can therefore write the equation as..

cxyx

dxxyx

xyxdxd

cos3

.sin

sin

2

2

2

xyx

cc

cos33

330

2

Substituting x=0, y=1

Solve the following differential equation…

cxyx

xyx

dxd

xyx

dxdy

yx

ln

1

12

2

2

2

2

yx

dxdy

yx

yx

dxd 2

2

22

When x=0 and y=1

xxy

xyx

c

ln1

1ln

1

2

2

When x=0 and y=1

You may notice that…

Exact derivativesWe were able to solve these two differential equations as the LHS were exact derivatives.

xxydxdyx sin322

xyx

dxdy

yx 12

2

2

Unfortunately this isn’t always the case…

xxy

dxdy 2

This is not an exact derivative of anything.What could I multiply it by to make it an exact derivative??22xyx

dxdy

We call “x” the integrating factor

Now solve this differential equation

xcxy

cxxy

xxy

xxydxd

xyxdxdy

32

32

2

2

2

2

3

2

2

2

Integrating FactorsIt is not always obvious what the integrating factor is but fortunately, we can work it out.

Consider the differential equation..

xQxyPdxdy

Notice the coefficient of dy/dx is 1

xQxfxPxyfdxdyxf )()()( We multiply by an

integrating factor, f(x)

xQxyPdxdy

Notice the coefficient of dy/dx is 1

xQxfxPxyfdxdyxf )()()( We multiply by an

integrating factor, f(x)

We multiply by an integrating factor, f(x) xQxfxPxyf

dxdyxf )()()(

xQxyPdxdy

Notice the coefficient of dy/dx is 1

The first term is one of the terms we get when we differentiate yf(x).

xyfdxdyxfxyf

dxd '

We want the second term to be the other.

)(')( xyfxPxyf

)()('xfxfxP

dxxPexf

xfdxxP

dxxfxfdxxP

)(

ln

)()('

The Shizzle

dxxP

exf)(

This is what we use to find the integrating factor

Example 1 dxxP

exf)(

23xey

dxdyx

x

Note: The coefficient of dy/dx must be 1

3

3xe

xy

dxdy x

P(x) is the coefficient of y

3lnln3

3

3

)(

3

xeexf

exf

xxP

xx

dxx

This is our integrating factor

x

x

eyxdxd

eyxdxdyx

3

23 3

3

3

xcey

cedxeyxx

xx

Example 2 dxxP

exf)(

ecxxydxdy coscot

xeexf xxdxsinsinlncot

This is our integrating factor

xcxy

cxdxxy

xydxd

xydxdyx

sin

1sin

1sin

1cossin

3

3

xcey

cedxeyxx

xx

Exercise

1. Find the general solution to the differential equation

2

1x

ydxdyx

2. Given that when x = 0, y = 1, solve the differential equation

xxeydxdy 4

Answers

1. Find the general solution to the differential equation

2

1x

ydxdyx

2. Given that when x = 0, y = 1, solve the differential equation

xxeydxdy 4

xcxy

21

xx eexy 212

Links to other resources

Mathsnet Q1Mathsnet Q2Mathsnet Q3Mathsnet Q4Mathsnet Q5