DIFFERENTIAL EQUATIONS

1. CHAPTER FIVE:THE INDEFINITE INTEGRAL AND BASIC DIFFERENTIAL EQUATIONSE-mail : vedicger108@hotmail.com

2. This chapter about the indefinite integral. In chapter 4, we dealt with definite integrals, in which we had limits to deal with. However, with indefinite integrals, we don’t have limits, i.e. we don’t have the a or b in the integral sign. • You are expected to know ALL the rules of differentiation you learned from chapter 4. • You will also deal with the infamous words “differential equations.” Differential equations (commonly referred as “diff-eq”s) basically an equation that has a derivative in it. More on that, when we get there. • For college calculus I students, this is the final chapter for you. You might go into a little bit of chapter 6, but that is very questionable. Start from the midpoint of this chapter, since you are not responsible for differential equations.

3. DIFFERENTIAL EQUATIONS • A differential equation is an equation that contains a derivative. For example, this is a differential equation. • From antidifferentiating skills from last chapter, we can solve this equation for y.

4. THE CONCEPT OF THE DIFFERENTIAL EQUATION • The dy/dx = f(x) means that f(x) is a rate. To solve a differential equation means to solve for the general solution. By integrating. It is more involved than just integrating. Let’s look at an example:

5. EXAMPLE 1 • GIVEN • Multiply both sides by dx to isolate dy. Bring the dx with the x and dy with the y. • Since you have the variable of integration attached, you are able to integrate both sides. Note: integral sign without limits means to merely find the antiderivative of that function • Notice on the right, there is a C. Constant of integration.

6. C?? What is that? • Remember from chapter 2? The derivative of a constant is 0. But when you integrate, you have to take into account that there is a possible constant involved. • Theoretically, a differential equation has infinite solutions. • To solve for C, you will receive an initial value problem which will give y(0) value. Then you can plug 0 in for x and the y(0) in for y. • Continuing the previous problem, let’s say that y(0)=2.

7. Solving for c.

8. SLOPE FIELDS***AP CALCULUS MATERIAL ONLY*** • We just solved for the differential equation analytically (‘algebraically’). The slope field (also known as vector field and directional field) will give us a qualitative analysis. • The graph shows all the possible slopes in the form of a field. • The arrows show the basic trend of how the slope changes. Using the initial condition, you can draw your solution. • For the previous example, the slope field will be very simple to draw.

9. SLOPE FIELD FOR EXAMPLE 1 • Notice how slope field TRACES the tangent lines of points from the antiderivative from various constants. • For the curve that is relevant with the correct, in our last problem C=2, connect those particular tangent lines and heavily bold it. • I drew this by hand, so please forgive my sloppiness with the slopes. _/\_ 

10. SLOPE FIELDS • The previous was so easy that a slope field was really not required. • However, there are many differential equations that will not yield easily to form such a slope field.

11. HOW TO DRAW SLOPE FIELDS • Consider dy/dx=-2xy. This is the formula for SLOPE • To find the slope, you need both an (x,y) coordinate. For example, if you use (1,-1), then the slope = (-2)(1)(-1)=2.

12. DRAWING SLOPE FIELDS • Start from (1,-1) and make a small line with the slope of 2. (Remember in high school, when you did lines, how did you do slope? Difference in y over difference in x). • Thus, the solution of the differential equation with the initial condition y(1)=-1 will look similar to this line segment as long as we stay close to x=-1.

13. DRAWING SLOPE FIELDS • However, simply drawing one line will not help us at all. You have to draw several lines. This what gets the Durvasa Muni out of the calculus students! • Then connect the “lines” horizontally to fit a curve amongst the tangent lines. These lines are formed from various C values.

14. SLOPE FIELDS • This topic of slope fields will be discussed highly in a college differential equations course. The AB Calculus exam, since 2002, has included slope fields in the curriculum, they have to know just as much about slope fields as BC Calculus. • The college calculus teachers generally like to skip over such topics of differential equations, even the easy ones like the first example. • Let’s consider the last example dy/dx=-2xy. Say we were the 2001 graduating class (that’s my graduating class ) and we didn’t learn slope fields. How would we such such an equation since there is a y there.

15. SEPERATION OF VARIABLES • Such equations are known as separable differential equations. The way to go about solving such equations (raksasas lol ) is to round up your y terms with dy and round up your x terms with dx. • When integrating dy/y, remember: the derivative of ln y is 1/y. Therefore the integral of 1/y is ln y.

16. INTIAL VALUE PROBLEM • Let’s say that y(1)=-1. We can find C that way. • And finally, your exact answer.

17. AUTONOMOUS SEPARABLE DIFFERENTIAL EQUATIONS (A.S.D.E.) • A differential equation that is autonomous means that the derivative does not depend on the independent variable. For example, The equation below is an autonomous equation. Notice that there is no x involved.

18. SOLVING A.S.D.E • You can still separate the y and bring dx to the right. • The process is the same. 

19. POPULATION GROWTH • The rate of the population for New Vrndavana is shown by the differential equation, dy/dt=ky. If k is the population constant, let k=1. If t is measure in years and if y(0)=200, then what is the predicted population in 2 years?

20. POPULATION GROWTH • Since we were given rate, we have to find the population by solving the differential equation. • You can use separation of variables to solve this autonomous equation. Note how I solve for y. • GIVEN: • Separate variables • Integrate • Exponentiate to undo the natural log on the left. • ekt and eC can be separated as such due to the laws of exponents. • eC itself is a constant so you can rename that number as C.

21. SOLVING FOR C. • Not too difficult, since the problem said earlier, that k=1, we don’t need to keep the k in there. Since, t=0, the exponent iteself will be zero, therefore e0=1, thus y(0)=C or C=200.

22. PREDICTING THE POPULATION • You can predict the population, by simply plugging t=2 in y(t) and get the answer. • 1477 people is the answer. • See how Lord Caitanya was true when He says, “Every town and village”! New Vrndavana population increases! Jaya! 

23. CONCLUSION TO DIFFERENTIAL EQUATION CONCEPTS • After seeing the population growth problem, it is best stop here for the differential equations portion of this chapter. This topic is covered in the AB and BC Calculus exams. • Knowing that the curriculum changes a lot each year, when necessary, I will change the presentations to fit their standards.

24. INDEFINITE INTEGRALS • CALCULUS 1 STUDENTS, YOU START THIS SECTION • An indefinite integral came up frequently in the differential equations part of the chapter. It is the integral with no limits. It is used to merely antidifferentiate functions.

25. INTEGRALS THAT SHOULD BE KNOWN

26. U-SUBSTITUTION • If you read the conclusion of Chapter 5, where I threw my opinions, you will notice how I described the reason why integral calculus is very long. The only reason why is because the integral of a product or quotient of two functions is not equal to the product or quotient of the integrals of the functions, respectively. • Therefore, a good amount integral calculus is antidifferentiation. There are many ways to integrate functions. If you are a calculus I or AB student, then this will be the only way you will learn this semester. If you are a BC student, you will learn many ways, this being your first way. • Again, memorize that table on the last slide with all the integrals!!!!!!

27. INTEGRATION BY SUBSTITUTION (“u” substitution) • Consider this function. You can use the binomial theorem, expand it and integrate each term piece by piece. Very tedious but doable.

28. U-SUBSTITUTION • However, there is no way you can break something like this down and integrate easily. Therefore, with the rules that we learned so far, you cannot integrate such a function.

29. SUBSTITUTING • This function could be a bit easier as well as pleasing to look at (but Krsna is most pleasing to look at), if you substituted a variable and integrated that way. Like the chain rule, let’s use the argument to be the “u.”

30. CHOOSING “U” • GIVEN INTEGRAL • Defined u(x). • Substituted u into the integral. • So we can integrate, right?? WRONG! Look at the variable of integration.. Its dx. We didn’t take into account this. Remember the Reimann sum? It was the function times the really small change in x, namely dx. Therefore, we have to find a dx substitute.

31. A dx substitute • To get dx, differentiate u, thus, you will get du/dx. Multiply both sides by dx. • Note, that we have 2dx = du. We don’t want a 2dx, we just want a dx. So we simply divide both sides by 2. • Then plug du/2 in for dx in the integral. Now integrate! Note, how I was able to pull the ½ out of the integral.

32. LET’S TRY SOMETHING MORE CHALLENGING • GIVEN: • Name u: • Notice that we solved for xdx. We gave a name for x2-3, but needed a name for xdx, therefore, we only solved for what we needed. This is a key idea in doing these u-substitution problems. Solve ONLY for what you need.

33. U-SUBSTITUTION • Let’s try another one. Matter of fact, let’s “invent” the integral of tan x! • You have to set it as sin x/cos x. • Let u = cos x. thus, -du = sin x. • DO NOT LET u = sin x in this example, because du = cos x dx, and we don’t have it. In fact, we have dx/cos x if you look at it. • Remember rules of logs: • -ln|a| = ln|1/a|

34. EVALUATING DEFINITE INTEGRALS USING u-substitution • In terms of integrating the function, itself, the rules are the same, however, there are two ways to evaluate it at the limits. • GIVEN:

35. DEFINITE INTEGRAL • Let’s worry about the function first. The expression under the square root looks like a good place to call that u. Another thing, I should have emphasized is that look for “derivative similarities.” You see x3 with a constant. You know that if you differentiate that, you’ll get just an x2. ALWAYS look for derivative similarity!

36. EVALUATING THE LIMITS • If you want to use u, then evaluate u at both limits. In this case u(0)=0 and u(2)=4. • You could also, after integrating the function with respect to u, you can replace the u(x) back into and then evaluate using the original limits

37. Why is there no +C in the definite integral? • If you look at the fundemental theorem of calculus, F(b)-F(a), you will see that both +C values cancel. It becomes immaterial whatever the function has a constant added to it or not. • Whenever you not definite (indefinite), then add Lord Caitanya (+C) to everything. Then you’ll have infinite answers !!!

38. SUMMARY for Diff-Eqs. • A differential equation is an equation that contains a derivative in it. There are infinite solutions, but with the help of an initial value you problem, you can solve for an arbitrary constant, to help you get the final equation. • You can solve it analytically using integration, or qualitatively by slope fields. • Slope fields (also known as vector or direction fields) show the general trend of solutions of the differential equation with various C values. • An autonomous equation is a differential equation which does not depend on the independent variable. For example, dy/dx=f(y) is autonomous. • Exponential growth can be identified with the equation f(t)=Cekt.