We Know, differential equations have different applications in real life such as the law of exponential change, radioactivity, the half-life of a radioactive element, carbon dating temperature, and rate of charge electric circuit.
What is a differential equation?
An equation involving independent and dependent variables and the derivative or differentials of one or more dependent Valuables with respect to one or more independent valuable Called differential equation or an equation containing derivatives are called Differential equations
Examples for differential equations
- dy / dx = sin x + cos x
- d2y / dx2 + dy/dx + y = 0
- d2y / dx2 = xy
- ∂y / ∂x = y
There are two main classes of differential equations
- Ordinary differential equations
- partial differential equations
Ordinary differential equations
Differential equations that involve derivatives with a single indigent variable are known as ordinary differential equations.
from the above example, 1, 2 & 3 are an ordinary differential equation
Partial differential equations
Differential equations contain two or more independent variables and partial derivatives with respect to them is called partial differentials
Example 4 is a partial differential equation
Order of a differential equation
The order of the highest order derivative involved in a differential equation is called the order of the differential equation
Examples 1 & 4 are 1st order and 2 & 3 are second-order differential equations
Degree of a differential equation
The degree of a differential equation is the degree of the highest order derivative of the equation.
The above example, all are first-degree differential equation
(dy/dx)4 = xy
(d2y/dx2)4 + dy/dx + y= 0
(d2y/dx2)4 = xy
Here 1 & 2 are 4th degree and 3rd is 2nd degree
Linear and nonlinear differential equation
A differential equation in which the dependent variable and all its derivatives are present occur in the 1st degree only and no products of dependent variables or derivatives occur known as the linear differential equation
A linear Differential equation is an equation that is not linear
Real life application of differential equations
- The law of exponential change
- The half-life of a radioactive element
- Radioactivity, carbon dating
- Temperature Rate of change
- Electric circuit
Application of differential equation in law of exponential change
Suppose “y” represents a quality (velocity, temperature, ….) That increases or decreases at a rate that at any time “t” is proportional to the amount present.
Suppose that “y” take the value y₀ at t = 0, there we can find “y” is a function of “t”
By solving the following initial value problem
dy/dt = ky
y = y0 when t = 0
y is positive and increasing then k is positive and the rate of growth is proportional to what has already been accumulated.
ln y = kt + lnC
y = eᵏᵗ × eᶜ
y = Aeᵏᵗ
To determine A we have to use the initial condition y=y₀ when t=0 putting the t=0 the above equation we get
y = y0 eᵏᵗ
From the above arrangement, it follows that the exponential change y = y0 eᵏᵗ
Is growth if k>0. and Is decay if k<0
In biology, it is often observed that the rate at which certain bacteria grow is proposition all to the number of bacterial presents at any time.
dy / dt = ky
(1/y)(dy / dt) = k
dy/y = k dt
When an atom emits some of its mass as radiation. A reminder of the atom reform to mack an atom of some new element.
This process of radiation and change is called radioactive decay and
An element whose atoms go spontaneously through this process is called radioactive.
The experiment has shown that at any given time the rate at which a radioactive element decays is approximately proportional to the number of radioactive nuclei present
Thus decay is described by the equation
dy / dt = -ky k<0
If y₀ is the number of radioactive nuclei present at time zero. Then the number of radioactive nuclei present after time “t”
Is, y = y₀e⁻ᵏᵗ k<0
The half-life of a radioactive element
The half-life of a radioactive element is the time required for half of the radioactive nuclei present in a sample to decay
Suppose y₀ is the number of radioactive nuclei initially present in the sample and the number still present at any later time t.y is the number of nuclei present equals half the original number y₀
y₀e⁻ᵏᵗ = y0 / 2
e⁻ᵏᵗ = 1/2
- kt = – ln 2
t = (ln 2)/k
half-life = (ln 2)/k
From the derivation of half-life, it follows that the half-life is a constant that does not depend on the number of radioactive nuclei present in the sample but only on the radioactive substance
Application of differential equations in radioactivity, carbon dating
The science of radio geology applies our knowledge of radioactivity to geology
It is known that uranium 238 undergoes radioactive decay with a half-life T = 4.55 billion years. During decay, it becomes radium 226
Similarly, C₁₄ is to radioactive carbon. Carbon in CO₂ changes to C₁₄ due to cosmic rays in the atmosphere. This Carbon dioxide is absorbed by plants. Also, radioactive carbon is accurate in animals by eating plants
In living cell rate of integration of C₁₄ exactly balanced with the rate of disinfection
So when an organisation die it use to ingest C₁₄, its C₁₄ carbon concentration decreases through the disintegration of C₁₄ present
This method used to estimate the age of organic materials and animal fossils also use to find the age of charcoal
Application of differential equations in temperature rate of change (Newton’s law of cooling)
Under certain conditions temperature rate of change of body is proportional to the difference between the temperature of the body and the temperature of the surrounding medium
dT / dt = k(T – T0)
dT / (T – T0) = k dt
log(T – T0) = kt + C
T – T0 = ekt + C
T = T0 + ekt + C
We can find the death time of a person with a temperature difference.
Normal body temperature is 37°C after death it decreases with time.
The temperature changes in the atmosphere are constant then easily find out the death time
Application of differential equations in electric circuit
In a series circuit containing a resistor and an inductor only Kirchoff’s second law state that the sum of the voltage drop across the inductor “L(di/dt)” and the voltage drop across the resistor “iR” is the same as the input voltage E(t) on the current(i) the differential equation is
L (di/dt) + iR = E(t)