Download Differential Equations

Lecture IV
The elements of
higher mathematics
Differential
Equations
Lecture questions
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Differential equation definition
Ordinary and partial differential equations
The order of differential equation
General and particular solutions of differential
equation
Particular solution, supplementary conditions
Separable differential equation
Differential Equation
• A differential equation is a mathematical
equation for an unknown function of one or
several variables that relates the values of
the function itself and its derivatives of
various orders or differentials.
Universality of mathematical
description
• The study of differential equations is a wide field
in pure and applied mathematics, physics,
meteorology, engineering, chemistry, biology
and economics. All of these disciplines are
concerned with differential equations of various
types. They are used to model the behavior of
complex systems. Many fundamental laws can
be formulated as differential equations.
Differential equations play an important role in
modeling virtually every physical, technical, or
biological process, from celestial motion, to
bridge design, to interactions between neurons.
Ordinary and partial differential
equations
• An ordinary differential equation (or ODE)
is a relation that contains functions of a single
independent variable, and one or more of their
derivatives with respect to that variable or
differentials.
• A partial differential equation (PDE) is a
differential equation in which the unknown
function is a function of multiple independent
variables and the equation involves its partial
derivatives or partial differentials.
Order
• The order of differential equation is
defined as the order of the highest
derivative of the dependent variable with
respect to the independent variable
y
appearing in the

x  yequation.
ln x
• For example :
is the firstorder ODE
y  y  0
•
is the second-order ODE
• To solve the equation means to
determine the unknown function which will
turn the equation into an identity upon
substitution.
(Solving
a
differential
equation means finding a function that
satisfies the given differential equation.)
• Solving a differential equation always
involves one or more integration steps.
General and Particular Solutions
• The general solution of the differential
equation y′ = 2 x is y = x2 + c, where c is any
arbitrary constant. Note that there are
actually infinitely many particular solutions,
such as y = x2 + 1, y = x2 − 7, or y = x2 + π,
since any constant c may be chosen.
• Geometrically, the differential equation y′ = 2
x says that at each point ( x, y) on some
curve y = y( x), the slope is equal to 2 x. The
solution obtained for the differential equation
shows that this property is satisfied by any
member of the family of curves y = x2 + c
Particular solutions graphs
Particular solution, supplementary
conditions
• If one particular solution (or integral curve)
is desired, the differential equation is
appended with one or more
supplementary conditions. These
additional conditions uniquely specify the
value of the arbitrary constant or constants
in the general solution.
Separable differential equation
is the differential equation of the
form
f ( x)u ( y )dx  v( x) ( y )dy  0
To solve a separable differential
equation perform the following
steps:
• 1) Separate variables x and y in different
sides of equation together with differentials
dx and dy. Move any term into the left side
v( xboth
)u ( yparts
)  0 by
and divide
to obtain the equation where the variables
are separated:
f ( x)u ( y)
v( x) ( y)
dx  
dy
v( x)u ( y)
v( x)u ( y)
f ( x)
 ( y)
dx  
dy
v( x)
u( y)
• 2) Integrate the equation: f ( x)
 ( y)
 v( x) dx   u( y) dy
• 3) And obtain a general solution:
once we find antiderivatives, we put the +C on
just one side (because two additive constants
can be absorbed into one).
g ( x )  h( y )  C
• 4) If you are given the initial (or boundary)
conditions use them to find the particular
solution.
Thank you for your attention !