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Mathematics for NST Part IA Cambridge University, 2012-2013

Notes for Ordinary Differential Equations General properties of ODEs: Ordinary differential equations (ODEs) relate the value of some variable

y

to the values

x .

of its ordinary derivatives with respect to one other variable

Order of differential equations: The order of the highest derivative of

y

with respect to

x : First-order ODE:

Second-order ODE:

Form of equation:

Form of equation:

'

n th-order ODE: Form of equation:

''

F ( y ( n) , y n-1 , ... , y , x )=0

F ( y , y , x )=0

F ( y , y , x ) =0

Form of solution:

Form of solution:

Form of solution:

f ( y , x ) =const.

f ( y , x ) =const.

f ( y , x ) =const.

Solutions and conditions: In the solutions of ODEs, an arbitrary constant of integration appears. Each value of this constant gives a different unique solution to the equation:

* The expression for the solution including the arbitrary constant is called the general solution

* If a condition is given to specify a particular value of the constant, the expression for this solution is called a unique solution Two types of condition are generally encountered:

*

An initial condition is the value of the function at

t=0 , when one of the variables

is time.

*

A boundary condition is the value of the function at

x=0 , when one of the

variables is position.

First-order ordinary differential equations: Each different type of first-order ODE requires its own method of solution, as follows. Directly-integrable, firstorder:

dy
=f (x ) dx

Method of solution: Integrate directly to produce

y=[?] f (x)dx

such that the

problem is reduced to finding the integral of a single-variable function. By the Fundamental Theorem of Calculus, the solution is:

y=F ( x ) +c Which is a family of solutions for which each member is

Mathematics for NST Part IA Cambridge University, 2012-2013 distinguished by a unique value of the parameter

Separable, first-order:

c .

Method of solution:

g( y ) to produce a directly-integrable ODE:

Multiply by

dy f ( x )
=
dx g ( y )

dy

[?] g ( y ) dx dx=[?] f ( x)dx [?] [?] g ( y)dy=[?] f (x )dx G ( y )=F ( x )+ c Which is a family of solutions for which each member is distinguished by a unique value of the parameter

Linear, first-order:

*

y

*

y

A function of a single variable is raised, for example

An operator

is linear if

c .

1 is the highest power to which

f ( y )=ax+ b or f ( y )=dy /dx+ cy +d

L is linear if it satisfies the condition that:

L ( ax+by ) =aL ( x ) +bL ( y ) , for constants a and b

*

The operator

L=d /dx

while the operator

*

L=

is linear when acting on integrable functions of

x The operator for linear first-order ODEs is:

d
+ p(x ) dx For some function

*

[?] dx

x ,

is linear when acting on differentiable functions of

p(x) of

x .

Linear first-order ODEs can be divided into two categories: homogeneous for

Ly=0

and inhomogeneous

Homogeneous, linear, firstorder:

dy
+ p ( x ) y=0 dx

for

Ly=f ( x) .

Method of solution: This is a separable first-order ODE:

dy dy
=-p ( x ) y [?] [?] =-[?] p ( x ) dx [?] ln y=-[?] p (x)dx dx y

- p (x ) dx
[?] y=e [?]

y=e-P (x )+c Which is a family of solutions for which each member is

Mathematics for NST Part IA Cambridge University, 2012-2013 distinguished by a unique value of the parameter

Inhomogeneous, linear, firstorder:

dy
+ p ( x ) y=f (x ) dx

c .

Method of solution: Reduce to a separable first-order ODE by multiplying by an integrating factor

m( x) :

dy
+ m ( x ) p ( x ) y =m ( x ) f (x ) dx

m (x )

m( x)

is chosen so that the left-hand side of this

x , such that the

equation is a derivative with respect to equation becomes separable

m(x)

therefore satisfies

the condition that:

m (x )

dy d d
= ( m ( x ) y )- y ( m ( x ) ) dx dx dx

[?] LHS=
[?]

d d d ( m ( x ) y ) - y ( m ( x ) ) + m ( x ) p ( x ) y [?] y ( m ( x ) )=m ( x ) p ( x ) y dx dx dx

d dm
( m ( x ) )=m ( x ) p ( x ) [?] =m ( x ) p ( x ) dx dx

[?][?]

dm
= p(x )dx
m [?]

x [?] m ( x )=e

[?] p ( x ' ) dx '

Method of solution: This integrating factor can then be substituted back into the original ODE to give:

m (x )

dy d
+ m ( x ) p ( x ) y =m ( x ) f ( x ) [?] ( ym ( x ) ) =m ( x ) f ( x ) dx dx

[?] ym ( x )=[?] m ( x ) f ( x ) dx y=

1 [?] m ( x ) f ( x ) dx
m (x )

Mathematics for NST Part IA Cambridge University, 2012-2013 Which is a family of solutions for which each member is distinguished by a unique value of the parameter

c

(not shown).

Solution spaces for linear, first-order ODEs: Homogeneous solution space: Linear, thus solutions satisfy the principle of superposition:

y1

For solutions

and

y2

:

Inhomogeneous solution space: Non-linear, thus solutions do not satisfy the principle of superposition:

d y1 d y2
+ p ( x ) y1 =0,
+ p ( x ) y 2=0 dx dx

a

d ( a y 1 +b y 2 ) dx

( ddxy + p ( x ) y )+ b( ddxy + p ( x) y ) 2

1 a

2 0+0=0 Thus

:

y=a y 1 +b y 2 :

+ p ( x ) ( a y 1 +b y 2 )

1 y2

A linear combination of these solutions is

y=a y 1 +b y 2 : dx

and

d y1 d y2
+ p ( x ) y1 =f ( x ),
+ p ( x ) y 2=f ( x) dx dx

A linear combination of these solutions is

d ( a y 1 +b y 2 )

y1

For solutions

+ p ( x ) ( a y 1 +b y 2 )

( ddxy + p ( x ) y )+ b( ddxy + p ( x) y ) 1

2 1

2 af (x )+ bf ( x)[?] f (x )

a y 1 +b y 2

is also a solution; the

Thus

solutions are linear and superimposable.

a y 1 +b y 2 is not a solution; the

solutions are not linear and superimposable.

Substitutions for first-order ODEs: Many non-linear or non-separable first-order ODEs can be reduced to linear or separable ODEs by means of a suitable substitution. The two most common types of first-order ODE which can be solved in this manner are homogeneous differential equations and Bernoulli differential equations: Homogeneous differential equations: Here 'homogeneous' does not refer to the property described above for linear first-order ODEs; instead, it refers to the property that if both

x and

y

are

scaled by the same constant

a ,

the equation remains unchanged:

dy y
=f dx x

()

Method of solution: Substitution function of

LHS=
[?]

y=ux

x :

d ( ux ) ux , RHS=f
=f (u) dx x

( )

du du f ( u ) -u x +u=f ( u ) [?] =
dx dx x

Which is a separable first-order ODE.

u is another

Mathematics for NST Part IA Cambridge University, 2012-2013 Bernoulli differential equations:

dy
+ p ( x ) y=q ( x ) y n dx When

n=0 , the differential

equation is linear, and when

Method of solution: The substitution

equation becomes a linear ODE in

separable, but can be made so by a suitable substitution.

z :

dz dz dy dy
=
=( 1-n ) y-n =( 1-n ) y -n (-p ( x ) y+ q ( x ) y n ) =( 1-n ) ( q ( x ) dx dy dx dx

n=1 , it is separable. For

general n , it is neither linear nor

z= y 1-n is made, for which the

[?]

dz
+ ( 1-n ) p ( x ) z =( 1-n ) q (z) dx

Which is an inhomogeneous linear first-order ODE, and can be solved via the use of an integrating factor.

Mathematics for NST Part IA Cambridge University, 2012-2013

Second-order ordinary differential equations: Second-order ODEs are generally more difficult to solve than first-order ODEs, and some of the simpler categories which can be identified for first-order ODEs, such as separable equations, do not extend to second-order ODEs. Here we will only examine linear secondorder ODEs, which are the most important time of second-order ODE, describing many physical phenomena:

d2 y dy
+ p ( x ) +q ( x ) y=f ( x) 2 dx dx d2 d
+ p ( x ) + q( x ) . In 2 dx dx

Which is linear in

y , and displays the differential operator

physical systems,

f ( x) is a 'forcing function' which describes the force applied to the

system. Like linear first-order ODEs, linear second-order ODEs can be homogeneous or inhomogeneous: Homogeneous/'unforced':

Inhomogeneous/'forced':

f ( x )=0

f ( x)[?] 0

Ly=0

Ly=f ( x)

Solutions: Obey the principle of superposition, such that

u( x ) and v (x ) are solutions to the

if

ODE, then any linear combination of these,

au+bv , is also a solution.

Solutions: Do not obey the principle of superposition, but are formed from the sum of two parts - the 'particular integral' (

y p ) and the

'complementary function' (

yc

).

yc

is

the solution to the corresponding homogeneous equation

Ly=0 , producing

the two required arbitrary constants, and thus

yp

is the solution to

Ly=f

which

produces no more arbitrary constants. The sum of these solutions satisfies the equation as follows:

L ( y p+ y c ) =L ( y p ) + L ( y c ) =f + 0=f Dimension of the solution space: Dimension of the solution space: The dimension of a solution space is equivalent to the number of linearly-independent vectors which span it, thus the number of free parameters in the general solution. An

n th order homogeneous linear

In addition to the homogeneous linear solution

y c , the inhomogeneous solution

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