UNIVERSITY OF TECHNOLOGY
MATERIAL ENGINEERING
DEPARTMENT
MATHEMATICS II
DR. KADHUM MUTTAR SHABEEB
First Semester 2011-2012
Ordinary Differential
Equations
Ordinary Differential Equations
By an ordinary differential equation we mean a relation which involves
one or several derivatives of an unspecified function y of x with
respect to x; the relation may also involve y itself, given function of x,
and constants.
For example:
y’ = cos x
y’’ + 4y =0
x2y’’’y’ + 2exy’’ = (x2 + 2) y2
Order of Differential Equations
An ordinary differential equation is said to be of order n if the nth
derivative of y with respect to x is the highest derivative of y in that
equation. For example:y’ = cos x
1st order
y’’ + 4y =0
2nd order
x2y’’’y’ + 2exy’’ = (x2 + 2) y
3rd order
Solution of Differential Eq.
A solution of differential Eq. is a differentiable function y=y(x)
defined on an interval I of x-values. That is, when y(x) and
its derivative are substituted into differential Eq., the
resulting equation is true for all x over the interval I.
Example:- verify that y = cos x is a solution of LJ͛͛н y = 0 for all
x.
Types of Solution
1- General Solution:- involves an arbitrary constant, for Ex.
y’=cos x
The general solution is
y = sin x + C
2- Particular Solution:- involves an specific constant, for Ex.
y’= x
when y(0) = 1
The particular solution is
y = (x2/2) + 1
H.W.
Verify that the given function is a solution for the given D.E.
a) y’ + y = x2 – 2
,
y = C e-x + x2 – 2x
b) Y’’’ = ex
,
y = ex + ax2 + bx + C
First Order Differential Equations
1) Separable :- is the differential Eq. that can be
reduced to the form:g(y) dy = f(x) dx
The solution is:ʃ g(y) dy = ʃ f(x) dx + C
Example:- Solve the following differential equations
a) 9yy’ + 4x = 0
b) y’ = -2xy
with initial value y(0) = 1
2) Equations Reducible to Separable Form
y
y  g ( )
x
• The general form
----------(1)
• This Eq. can be reduced to separable form as
follow:y
u

• Assume
-----------(2)
x
• Then y = ux and y’ = u’x + u --------------(3)
• Substitute Eq. (2) & (3) in (1), we get the
general form of solution
du
dx

g (u )  u x
Example:- solve
a) 2xyy’ – y2 + x2 = 0
b) y’ = (y-x)/(y+x)
3) Exact
The differential Eq. of the form
M(x,y) dx + N(x,y) dy = 0
M N
is exact if and only if

y
x
The solution of exact differential Eq. is
 M ( x, y)dx   N ( y)dy  C
Example:- Solve
a) (x3+3xy) dx + (3x2y+y3) dy = 0
b) 2x sin 3y dx + 3x2cos3y dy + 0
• Reduction to Exact form
Sometimes a given differential Eq.
P(x,y) dx + Q(x,y) dy = 0 -------------(1)
is not exact but can be made exact by multiplying
it by integrating factor F.
Eq. (1) becomes exact
FP(x,y) dx + FQ(x,y) dy = 0 -------------(2)
F ( x)  exp  R( x)dx
1 P Q
R( x)  ( 
)
Q y x
Example:- Solve 2 sin(y2) dx + xy cos(y2) dy = 0
4) Linear Equations
A first-order linear differential equation is one that can be
written in the form y’ + P(x) y = Q(x)
where P and Q are continuous functions of x
• To solve the linear equation, multiply both sides by the
integrating factor I(x)=e∫p(x)dx and integrate both sides.
Example:- Solve xy’ = x2 + 3y
5) Bernoulli Equation
• General form y’ + p(x) y = Q(x) yn
• This D.E. can be reduced to linear D.E. by
multiplying it by y-n and assume z=y1-n
• Then differentiate z with respect to x and
substitute z and its derivative in above Eq.
Example:- Solve y’ + xy = xy2