Lecture 2 Ordinary Differential Equations fall semester

1. Differential Equations Lecture 2Ordinary Differential Equationsfall semester Instructor: A. S. Brwa / MSc. In Structural Engineering College of Engineering / Ishik University

2. Ishik University Ordinary Differential Equations Ordinary Differential Equation (ODE): is a differential equation containing one or more functions of one independent variable and its derivativeses Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

3. Ishik University Learning Outcomes Understand the use of differential equations in modelling engineering systems Identify the order and type of a differential equation recognize the nature of a general solution Determine the nature of the appropriate additional conditions which will give a unique solution to the equation Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

4. Ishik University Ordinary Differential Equations Rate of Change: A rate of change is a rate that describes how one quantity changes in relation to another quantity. If xis the independent variable and y is the dependent variable, then Rates of change can be positive or negative. This corresponds to an increase or decrease in the y –value between the two data points. When a quantity does not change over time, it is called zero rate of change Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

5. Ishik University Ordinary Differential Equations Many models of engineering systems involve the rate of change of a quantity. There is thus a need to incorporate derivatives into the mathematical model. These mathematical models are examples of differential equations. Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

6. Ishik University Ordinary Differential Equations Accompanying the differential equation will be one or more conditions that let us obtain a unique solution to a particular problem. Often we solve the differential equation first to obtain a general solution; then we apply the conditions to obtain the unique solution. It is important to know which conditions must be specified in order to obtain a unique solution. Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

7. Ishik University Case Study Newton’s law of cooling as an example When a hot liquid is placed in a cooler environment, experimental observation shows that its temperature decreases to approximately that of its surroundings. A typical graph of the temperature of the liquid plotted against time is shown in Figure 1. Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

8. Ishik University Faculty of Engineering – Differential Equations – Lecture 2 – 1st Order ODE

9. Ishik University Case Study After an initially rapid decrease the temperature changes progressively less rapidly and eventually the curve appears to ‘flatten out’. Newton’s law of cooling states that the rate of cooling of liquid is proportional to the difference between its temperature and the temperature of its environment (the ambient temperature). Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

10. Ishik University Case Study To convert this into mathematics, let t be the time elapsed (in seconds, s), ythe temperature of the liquid (), and the temperature of the liquid at the start (t = 0). The temperature of the surroundings is denoted by. Now, we can create a mathematical equation which is equivalent to Newton’s law of coolingand state the accompanying condition. Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

11. Ishik University Case Study First, find an expression for the rate of cooling, and an expression for the difference between the liquid’s temperature and that of the environment: Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

12. Ishik University Case Study The rate of cooling is the rate of change of temperature with time: The temperature difference is Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

13. Ishik University Case Study Now formulate Newton’s law of cooling: You should obtain equivalently: dθdt = −k(θ − θs). Which is equivalent to is a positive constant of proportion Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

14. Ishik University Case Study The negativesign ok is present because is positive, whereas must be negative, since decreases with time. The units of are. The accompanying condition is which simply states the temperature of the liquid when the cooling begins. at t = 0 Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

15. Ishik University Case Study In the above Task we call t the independent variable and the dependent variable. Since the condition is given at t = 0. so, this is considered as an initial condition. For future reference, the solution of the above differential equation which satisfies the initial condition is. Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

16. Ishik University General Solution of a ODE The general solution of a ODE Consider the equation where A is an arbitrary constant. If we differentiate it we obtain and so, since we obtain An arbitrary constant is a constant whose value could be assumed to be anything, it doesn't depend on the other variables in an equation or expression. A constant that's not arbitrary can usually just take one value, such as 1, 2, or 10 Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

17. Ishik University General Solution of a ODE Thus a differential equation satisfied by y is Note that we have eliminated the arbitrary constant. Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

18. Ishik University General Solution of a ODE Now consider the equation Where, A and B are arbitrary constants. Differentiate once , we obtain Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

19. Ishik University General Solution of a ODE Differentiating a second time gives The right hand side is simply (−9) times the expression for y. Hence y satisfies the differential equation Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

20. Ishik University General Solution of a ODE Find a differential equation satisfied by Where, A and B are arbitrary constants. Differentiating once we obtain Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

21. Ishik University General Solution of a ODE Differentiating a second time we obtain Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

22. Ishik University General Solution of a ODE We have seen that an expression including one arbitrary constant required one differentiation to obtain a differential equation which eliminated the arbitrary constant. Where two constants were presented, two differentiations were required. Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

23. Ishik University General Solution of a ODE Integrate twice the differential equation Where, w and Lare constants, to find a general solution for y. Integrating once: where A is an arbitrary constant (of integration) Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

24. Ishik University General Solution of a ODE Integrating again: Where, B is second arbitrary constant (of integration) Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

25. Ishik University General Solution of a ODE Consider the simple differential equation On integrating, we obtain the general solution Where, C is an arbitrary constant. As C varies we get different solutions, each of which belongs to the family of solutions Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

26. Ishik University General Solution of a ODE Consider the simple differential equation On integrating, we obtain the general solution Where, C is an arbitrary constant. As C varies we get different solutions, each of which belongs to the family of solutions Figure-2 Shows different values of C Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations

27. Ishik University Home Work Find differential equations satisfied by Solve the differential equation Faculty of Engineering – Differential Equations – Lecture 2 – Ordinary Differential Equations