application of differential equations
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This document provides an introduction to differential equations and their applications. It discusses the history of differential equations, types of differential equations including ordinary differential equations (ODEs) and partial differential equations (PDEs). Examples of first order ODE applications given include Newton's Law of Cooling, electrical circuits, and population growth modeling. Mechanical oscillation modeling is also discussed. The document concludes that differential equations have wide applications in fields like rocket science, economics, and gaming.
application of differential equations
- 2. BY G.V.MANISH REDDY II B.Sc. (M.S.Cs) APPLICATION OF DIFFERENTIAL EQUATIONS
- 3. CONTENT Introduction History of Differential Equations Types of Differential Equations Application of Differential Equations
- 4. INTRODUCTION A Differential Equation is an equation containing the derivative of one or more dependent variables with respect to one or more independent variables.
- 5. INTRODUCTION For Example: 𝑥 ⅆ𝑦 ⅆ𝑥 = 𝑦 − 1 ⅆ𝑦 ⅆ𝑥 = 2𝑥𝑦 𝜕𝑥 𝜕𝑦 + 𝜕𝑦 𝜕𝑥 +2=0
- 6. Differential equation was part of calculus, which itself was independently invented by, English physicist Isaac Newton and German mathematician Gottfried Leibniz. History of Differential Equations
- 7. History of Differential Equations “Differential equations began with Leibniz, the Bernoulli brothers, and others from the 1680s, Not long after Newton’s ‘fluxional equations’ in the 1670s.”
- 8. TYPES OF DIFFERENTIAL EQUATIONS ODE (ORDINARY DIFFERENTIAL EQUATION) AND TYPES OF ORDINARY DIFFERENTIAL EQUATION 1 PDE (PARTIAL DIFFERENTIAL EQUATION) AND TYPES OF PARTIAL DIFFERENTIAL EQUATION 2
- 9. ORDINARY DIFFERENTIAL EQUATION (ODE) An equation contains only ordinary derivatives of one or more dependent variables of a single independent variable. For Example, dy/dx + 5y = ex , (dx/dt) + (dy/dt) = 2x + y
- 10. TYPES OF ORDINARY DIFFERENTIAL EQUATION I) FIRST ORDER ODE II) SECOND ORDER ODE III) HIGHER ORDER ODE
- 11. PARTIAL DIFFERENTIAL EQUATION (PDE) A differential equation involving partial derivatives of a dependent variable(one or more) with more than one independent variable is called a partial differential equation, hereafter denoted as PDE.
- 14. APPLICATION OF ODE MODELLING WITH FIRST-ORDER EQUATIONS Newton’s Law of Cooling Electrical Circuits MODELLING FREE MECHANICAL OSCILLATIONS No Damping Light Damping Heavy Damping MODELLING FORCED MECHANICAL OSCILLATIONS
- 15. NEWTON’S LAW OF COOLING Newton’s empirical law of cooling of an object in given by the linear first-order differential equation 𝑑𝑇 𝑑𝑡 = 𝛼(𝑇 − 𝑇 𝑚) This is a separable differential equation. We have 𝑑𝑇 (𝑇−𝑇 𝑚) = 𝛼𝑑𝑡 or ln|T-Tm|=t+c1 or T(t) = Tm+c2et
- 16. NEWTON’S LAW OF COOLING When a chicken is removed from an oven, its temperature is measured at 3000F. Three minutes later its temperature is 200o F. How long will it take for the chicken to cool off to a room temperature of 70oF?
- 17. NEWTON’S LAW OF COOLING Solution: In (4.1) we put Tm = 70 and T=300 at for t=0. T(0)=300=70+c2e.0 This gives c2=230 For t=3, T(3)=200 Now we put t=3, T(3)=200 and c2=230 in (4.1) then 200=70 + 230e.3 Or 𝑒3𝛼 = 130 230 Or 3𝛼 = 𝐼𝑛 13 23 𝛼 = 1 3 In 13 23 = −0.19018
- 18. NEWTON’S LAW OF COOLING Thus T(t)=70+230 e-0.19018t (4.2) We observe that (4.2) furnishes no finite solution to T(t)=70 since limit T(t) =70. t T(min) T(t) 20.1 75º 21.3 74º 22.8 73º 24.9 72º 28.6 71º 32.3 700
- 19. POPULATION GROWTH Finding population growth using differential equations. The equation to find is, 𝑑𝑁 𝑑𝑡 = kN, for k = (r − m). Note , N(t) = N0ekt = N0e(r−m)t
- 20. POPULATION GROWTH For example, Given the initial condition (IC) N(0) = 6 billion, determine the size of the human population in 100 years that our model predicts?
- 21. POPULATION GROWTH Solution: We have that at time t = 0, N(0) = N0 = 6 billion. Then in billions, N(t)=6e0.0125t so that when t = 100 we would have N(100) = 6e0.0125·100 = 6e1.25 = 6 · 3.49 = 20.94 Thus, with population around the 6 billion now, we should see about 21 billion people on Earth in 100 years based on the uncontrolled continuous growth model discussed here.
- 23. CONCLUSION It is also used in many fields like, Rocket science Economics It is useful in gaming also.
- 24. - MANISH