Applications of differential equation

Applications of
Differential
equation

Contents
– Introduction to differential equation
– Definition and types of differential equation
– Application of differential equation in real life
– Solved examples

– A Differential Equation is a mathematical equation that relates some
functions with it’s derivatives. Differential equations play a prominent
role in many disciplines including engineering, physics, economics and
biology.
– In biology and economics, differential equations are used to model the
behavior of complex systems.
– Many fundamental laws of physics and chemistry can be formulated as
differential equations.
– In mathematics, differential equations are studied from several different
perspectives, mostly concerned with their solutions, the set of functions
that satisfy the equation.

Differential Equation:
An equation involving derivatives of one or more dependent
variable with respect to one or more independent variable is
called a Differential Equation.
For example
ⅆ 𝒚
ⅆ𝒙
= 𝝀 𝒚
is differential equation where λ is a constant ,x is an
independent variable and y is a dependent variable.
Differential Equation are of two types
1.Ordinary differential equation
2. Partial differential equation

Ordinary differential equation
A differential equation involving derivatives with
respect to a single independent variable is called an
ordinary differential equation.
For example,
ⅆ𝒚
ⅆ𝒙
= 𝟐 𝐬𝐢𝐧 𝒙 , 𝒙 ∈ 𝟎, 𝒂 , 𝒂 > 𝟎
is a ordinary differential equation where x is
independent variable , and y is dependent variable .

Partial differential equation:
A differential equation involving derivatives with
respect to more than oneindependent variable is called
an partial differential equation.
For example,
𝝏 𝟐 𝒖
𝝏𝒙 𝟐
+
𝝏 𝟐 𝒖
𝝏𝒚 𝟐
= 𝟎 𝒘𝒉𝒆𝒓𝒆 𝒖 = 𝒖 𝒙, 𝒚 𝒂𝒏ⅆ 𝒙, 𝒚 𝝐𝜴
= 𝟏, 𝟎 × 𝟎, 𝟏
Is a partial differential equation where the domain of
definition Ω is the open unit rectangle of dimension 2.

Order:
The order of a differential equation is
the order of the highest order
derivativeoccurring in it. For
example
,
ⅆ2
𝑦
ⅆ𝑥2
+ 5
ⅆ𝑦
ⅆ𝑥
3
− 4𝑦 = ⅇ 𝑥
is the second order ordinary
differential equation.

Degree:
The degree of a differential equation is the degree
of the highest order derivativewhich occurs in it,
after the differential equation has been made free
from radicals andfractions as far as the derivatives
are concerned.
For example,
ⅆ3
𝑦
ⅆ𝑥3
+
ⅆ2
𝑦
ⅆ𝑥2
3
+
ⅆ𝑦
ⅆ𝑥
5
+ 𝑦 = 7
Is second order differential equation of degree one.

Application of
differential
equation in real life

Let P(t) be a quantity that increases with time t and the rate of increase is proportional to the
same quantity P as follows
d P / d t = k P
where d p / d t is the first derivative of P, k > 0 and t is the time.
The solution to the above first order differential equation is given by
P(t) = A ek t
where A is a constant not equal to 0.
If P = P0 at t = 0, then
P0 = A e0
which gives A = P0
The final form of the solution is given by
P(t) = P0 ek t
Assuming P0 is positive and since k is positive, P(t) is an increasing exponential. d P / d t = k P
is also called an exponential growth model.
1 : Exponential Growth - Population

Let M(t) be the amount of a product that decreases with time t and the rate of decrease is
proportional to the amount M as follows
d M / d t = - k M
where d M / d t is the first derivative of M, k > 0 and t is the time.
Solving the above first order differential equation we obtain
M(t) = A e- k t
where A is non zero constant.
It we assume that M = M0 at t = 0, then
M0 = A e0
which gives A = M0
The solution may be written as follows
M(t) = M0 e- k t
Assuming M0 is positive and since k is positive, M(t) is an decreasing exponential. d M / d
t = - k M is also called an exponential decay model.
2 : Exponential Decay - Radioactive Material

An object is dropped from a height at time t = 0. If h(t) is the height of the object at time
t, a(t) the acceleration and v(t) the velocity. The relationships between a, v and h are as
follows:
a(t) = dv / dt , v(t) = dh / dt.
For a falling object, a(t) is constant and is equal to g = -9.8 m/s.
Combining the above differential equations, we can easily deduce the following
equation
d 2h / dt 2 = g
Integrate both sides of the above equation to obtain
dh / dt = g t + v0
Integrate one more time to obtain
h(t) = (1/2) g t2 + v0 t + h0
The above equation describes the height of a falling object, from an initial height h0 at an
initial velocity v0, as a function of time.
3 : Falling Object

It is a model that describes, mathematically, the
change in temperature of an object in a given
environment. The law states that the rate of
change (in time) of the temperature is
proportional to the difference between the
temperature T of the object and the temperature
Te of the environment surrounding the object.
d T / d t = - k (T - Te)
4 : Newton's Law of Cooling

Let x = T - Te so that dx / dt = dT / dt
Using the above change of variable, the above differential equation becomes
d x / d t = - k x
The solution to the above differential equation is given by
x = A e - k t
substitute x by T - Te
T - Te = A e - k t
Assume that at t = 0 the temperature T = To
To - Te = A e 0
which gives A = To - Te
The final expression for T(t) i given by
T(t) = Te + (To - Te)e - k t
This last expression shows how the temperature T of the object changes with time.

5 : RL circuit
Let us consider the RL (resistor R and inductor L) circuit
shown above. At t = 0 the switch is closed and current
passes through the circuit. Electricity laws state that the
voltage across a resistor of resistance R is equal to R i and
the voltage across an inductor L is given by L di/dt (i is
the current). Another law gives an equation relating all
voltages in the above circuit as follows:
L di/dt + Ri = E , where E is a constant voltage.

Let us solve the above differential equation which may be written as follows
L [ di / dt ] / [E - R i] = 1
which may be written as
- (L / R) [ - R d i ] / [E - Ri] = dt
Integrate both sides
- (L / R) ln(E - R i) = t + c , c constant of integration.
Find constant c by setting i = 0 at t = 0 (when switch is closed) which gives
c = (-L / R) ln(E)
Substitute c in the solution
- (L / R) ln(E - R i) = t + (-L/R) ln (E)

which may be written
(L/R) ln (E)- (L / R) ln(E - R i) = t
ln[E/(E - Ri)] = t(R/L)
Change into exponential form
[E/(E - Ri)] = e
t(R/L)
Solving for i we obtain
i = (E/R) (1-e-Rt/L)
The starting model for the circuit is a differential equation
which when solved, gives an expression of the current in the
circuit as a function of time.

Applications of differential equation

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