Introduction to differential equation
Introduction to Mathematics
of Finance
Session 24
Introduction to differential equation
April 17, 2018
Introduction to differential equation
Outline
1 Introduction to differential equation
Introduction to differential equation
Differential equation
An equation containing the derivatives of one or
more dependent variables, with respect to one
or more independent variables, is said to be a
differential equation.
Examples
dy
+ 5y = ex
dx
d 2y dy
− + 6y = 0
dx 2 dx
dx dy
+ = 2x + y
dt dt
Introduction to differential equation
Partial differential equation
An equation involving partial derivatives of one
or more dependent variables of two or more
independent variables is called a partial
differential equation.
∂ 2u ∂ 2u
+ =0
∂x 2 ∂y 2
∂ 2u ∂ 2u ∂u
= − 2
∂x 2 ∂t 2 ∂t
Introduction to differential equation
Classification by order
The order of a differential equation is the order
of the highest derivative in the equation
3
d 2y dy
2
+ 5 − 4y = ex
dx dx
The above equation is a second-order
differential equation
Introduction to differential equation
General notation
In symbols we can express an nth-order
ordinary differential equation in one dependent
variable by the general form
0
F (x, y , y , · · · , y (n) ) = 0 (1)
where F is a real valued function of n + 2
0
variables: x, y , y , · · · , y (n) . This is equivalent to
d ny 0
n
= f (x, y , y , · · · , y (n−1) ) (2)
dx
where f is a real valued continuous function.
Introduction to differential equation
Classification by linearity
An nth-order ordinary differential equation is said to be
0
linear if F is linear in y , y , · · · , y (n) . This means that an
nth-order ODE is linear when (1) is
0
an (x)y (n) + an−1 (x)y (n−1) + · · · a1 (x)y + a0 (x)y = g(x)
That is
The dependent variable y and all its derivatives are
of the first order (power of each term is 1)
The coefficients a0 , a1 , · · · , an depend at most on the
independent variable x
Introduction to differential equation
Examples
(y − x)dx + 4ydy = 0
00 0
y − 2y + y = 0
3
d y dy
+ x − 5y = ex
dx 3 dx
are, in turn, linear first-, second-, and third-order ordinary
differential equations
0 d 2y d 4y
(1 − y )y + 2y = ex , + sin y = 0, + y2 = 0
dx 2 dx 4
are examples of nonlinear first-, second-, and fourth order
ODE.
Introduction to differential equation
Solution of an ODE
Any function φ, defined on an interval I and
possessing at least n derivatives that are
continuous on I, which when substituted into an
nth-order ordinary differential equation reduces
the equation to an identity, is said to be a
solution of the equation on the intereval
Introduction to differential equation
Example
1 4 dy
Verify that y = 16 x is a solution to dx = xy 1/2
on the interval (−∞, ∞)
Introduction to differential equation
Solving differential equations
Review material
Basic integration formulae
techniques of integration (integration by parts
and partial fraction decomposition)
Introduction to differential equation
Separable equation
A first order differential equation of the form
dy
= g(x)h(y )
dx
is said to be separable or to have separable
variables
Example:
dy dy
= y 2 xe3x+4y = y + sin x
dx dx
Introduction to differential equation
Method of solution
Observe that we can write the equation
dy
= g(x)h(y )
dx
as
1
dy = g(x)dx
h(y )
Taking integral on both sides give
H(y ) = G(x) + c
where H(y ) and G(x) are antiderivatives of
1/h(y ) and g(x) respectively.
Introduction to differential equation
Examples
Solve
(1 + x)dy − ydx = 0
dy x
dx = − y , where y (4) = −3
Introduction to differential equation
Linear equation
A first order differential equation of the form
dy
a1 (x) + a0 (x)y = g(x)
dx
is said to be a linear equation in the dependent
variable.
When g(x) = 0, the linear equation is said to be
homogenous; otherwise, it is nonhomogeneous.
Introduction to differential equation
Standard form
A more useful form, the standard form, of a
linear equation is
dy
+ P(x)y = f (x)
dx
We seek solution of the above equation
Introduction to differential equation
Method of solution
1
Put a linear equation into the standard form
2
From the standard form identify RP(x) and
then find the integrating factor e P(x)dx
3
Multiply the standard form of the equation
by the integrating factor. The left hand side
of the resulting equation is automatically the
derivative of the integrating factor and y :
d h P(x)dx i R
e y = e P(x)dx f (x)
dx
4
Integrate both sides of this last equation
Introduction to differential equation
Example
Solve
dy
dx − 3y = 0
dy
dx − 3y = 6
Introduction to differential equation
Class work
Solve:
dy
= e3x+2y
dx
dy
+ y = x, y (0) = 4
dx
