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Application of Differential
Equation
Presenting by Group Abacus
Welcome to our
Presentation
Group Members:
 Yeasin Hossain Emran ID: 162-15-1037
 Mohammad Salim Hosen ID: 162-15-1044
 MD. Rifat Rahman ID: 162-15-1049
 MD.Hashibur Rahman Anik ID: 162-15-1032
About 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 (DE).
 There are two types of differential equation.
 Ordinary Differential Equation.
 Partial Differential Equation.
 A differential equation is written in the form:
𝑑𝑦
𝑑𝑥
= … …….
Applications of Differential Equations
 Differential equations have a remarkable ability to predict the world
around us. They are used in a wide variety of disciplines, from biology,
economics, physics, chemistry and engineering.
 Some Applications of Differential Equations in real life are,
 In Population Models
 In Newtons law of cooling
 In radioactive decay law
 In molecular biology
 In Computer Science
Application 1 : Exponential Growth - Population
 One of the most basic examples of differential equations is the Malthusian
Law of population growth
𝑑𝑝
𝑑𝑡
= 𝑟𝑝 shows how the population (p) changes
with respect to time. The constant r will change depending on the
species.
 More complicated differential equations can be used to model the
relationship between predators and prey. For example, as predators
increase then prey decrease as more get eaten. But then the predators
will have less to eat and start to die out, which allows more prey to
survive. The interactions between the two populations are connected by
differential equations.
Graph of Growth Population
 First Graph shows when the predator is both very aggressive it will attack the
prey and rapidly eat the prey population, growing rapidly – before it runs out of
prey to eat and then there will be no other food, thus dying off again.
 Then predators will be less aggressive and it will lead to both populations in a
stable position (Graph two).
Application 2 : Newton's Law of Cooling
 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.
 Equation is:
𝑑𝑇
𝑑𝑡
= −𝑘(𝑇 = −𝑇𝑒)
If the above equation is solved we get, T(t) = Te + (To - Te)e - k t express how the
temperature T of the object changes with time.
Example of Newtons cooling law
 Forensics experts use Newton's Law of Cooling to find out when victims of
crimes died. They take the temperature of the body when they find it, and by
knowing that the average temperature of the human body is 98.6 degrees
initially and measuring the room temperature, they can find k and then find t.
Application 3 : Exponential Decay - Radioactive Material
 We can find radioactive material half life by solving differential equation.
 Let M be the amount of a product that decreases with time t and the rate
of decrease is proportional to the amount M as follows,
dM / d t = - kM
 where k > 0 and t is the time.
 Solving the above first order differential equation we obtain M = A e- k t
 where A is non zero constant.
Application 4 : In molecular biology
 Differential equations are of basic importance in molecular biology because
many biological laws and relations appear mathematically in the form of a
differential equation.
 The vast majority of quantitative models in cell and molecular biology are
formulated in terms of ordinary differential equations.
 Mathematical cell biology is a very active and fast growing interdisciplinary
area in which mathematical concepts, techniques, and models are applied to
a variety of problems in developmental medicine and bioengineering.
Application 5 : In Computer Science
 Computer applications are involved in several aspects such as modeling
underlying logic or complex fluid flow, machine learning or financial analysis.
 Differential equation are greatly used in game development
 For example,
 In a simple video game involving a jumping motion, a differential equation
is used to model the velocity of a character after the command is given to
return them to the ground in a simulated gravitational field.
Some Other Applications of Differential Equations
 Some Other Applications of Differential Equations are,
1) In medicine for modelling cancer growth or the spread of disease
2) In engineering for describing the movement of electricity
3) In chemistry for modelling chemical reactions
4) In economics to find optimum investment strategies
5) In physics to describe the motion of waves, pendulums or chaotic systems.
THANK YOU

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Application of Differential Equation

  • 3. Group Members:  Yeasin Hossain Emran ID: 162-15-1037  Mohammad Salim Hosen ID: 162-15-1044  MD. Rifat Rahman ID: 162-15-1049  MD.Hashibur Rahman Anik ID: 162-15-1032
  • 4. About 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 (DE).  There are two types of differential equation.  Ordinary Differential Equation.  Partial Differential Equation.  A differential equation is written in the form: 𝑑𝑦 𝑑𝑥 = … …….
  • 5. Applications of Differential Equations  Differential equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering.  Some Applications of Differential Equations in real life are,  In Population Models  In Newtons law of cooling  In radioactive decay law  In molecular biology  In Computer Science
  • 6. Application 1 : Exponential Growth - Population  One of the most basic examples of differential equations is the Malthusian Law of population growth 𝑑𝑝 𝑑𝑡 = 𝑟𝑝 shows how the population (p) changes with respect to time. The constant r will change depending on the species.  More complicated differential equations can be used to model the relationship between predators and prey. For example, as predators increase then prey decrease as more get eaten. But then the predators will have less to eat and start to die out, which allows more prey to survive. The interactions between the two populations are connected by differential equations.
  • 7. Graph of Growth Population  First Graph shows when the predator is both very aggressive it will attack the prey and rapidly eat the prey population, growing rapidly – before it runs out of prey to eat and then there will be no other food, thus dying off again.  Then predators will be less aggressive and it will lead to both populations in a stable position (Graph two).
  • 8. Application 2 : Newton's Law of Cooling  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.  Equation is: 𝑑𝑇 𝑑𝑡 = −𝑘(𝑇 = −𝑇𝑒) If the above equation is solved we get, T(t) = Te + (To - Te)e - k t express how the temperature T of the object changes with time.
  • 9. Example of Newtons cooling law  Forensics experts use Newton's Law of Cooling to find out when victims of crimes died. They take the temperature of the body when they find it, and by knowing that the average temperature of the human body is 98.6 degrees initially and measuring the room temperature, they can find k and then find t.
  • 10. Application 3 : Exponential Decay - Radioactive Material  We can find radioactive material half life by solving differential equation.  Let M be the amount of a product that decreases with time t and the rate of decrease is proportional to the amount M as follows, dM / d t = - kM  where k > 0 and t is the time.  Solving the above first order differential equation we obtain M = A e- k t  where A is non zero constant.
  • 11. Application 4 : In molecular biology  Differential equations are of basic importance in molecular biology because many biological laws and relations appear mathematically in the form of a differential equation.  The vast majority of quantitative models in cell and molecular biology are formulated in terms of ordinary differential equations.  Mathematical cell biology is a very active and fast growing interdisciplinary area in which mathematical concepts, techniques, and models are applied to a variety of problems in developmental medicine and bioengineering.
  • 12. Application 5 : In Computer Science  Computer applications are involved in several aspects such as modeling underlying logic or complex fluid flow, machine learning or financial analysis.  Differential equation are greatly used in game development  For example,  In a simple video game involving a jumping motion, a differential equation is used to model the velocity of a character after the command is given to return them to the ground in a simulated gravitational field.
  • 13. Some Other Applications of Differential Equations  Some Other Applications of Differential Equations are, 1) In medicine for modelling cancer growth or the spread of disease 2) In engineering for describing the movement of electricity 3) In chemistry for modelling chemical reactions 4) In economics to find optimum investment strategies 5) In physics to describe the motion of waves, pendulums or chaotic systems.