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DIFFERENTIAL EQUATIONS PowerPoint Presentation

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DIFFERENTIAL EQUATIONS

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DIFFERENTIAL EQUATIONS

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  1. 9 DIFFERENTIAL EQUATIONS

  2. DIFFERENTIAL EQUATIONS • We have looked at first-lodge differential equations from a geometric point of view (direction fields) and from a numerical indicate of view (Euler's method). • What about the symbolic signal of view?

  3. DIFFERENTIAL EQUATIONS • It would exist overnice to have an explicit formula for a solution of a differential equation. • Unfortunately, that is non always possible.

  4. DIFFERENTIAL EQUATIONS 9.3Separable Equations • In this section, nosotros will larn near: • Sure differential equations • that can be solved explicitly.

  5. SEPARABLE EQUATION • A separable equationis a commencement-order differential equation in which the expression for dy/dx tin can exist factored equally a function of x times a function of y. • In other words, information technology can be written in the form

  6. SEPARABLE EQUATIONS • The name separablecomes from the fact that the expression on the correct side tin can be "separated" into a office of xand a office of y.

  7. SEPARABLE EQUATIONS Equation 1 • Equivalently, if f(y) ≠ 0, we could write where

  8. SEPARABLE EQUATIONS • To solve this equation, we rewrite information technology in the differential form • h(y) dy = m(x) dxso that: • All y'due south are on one side of the equation. • All x'due south are on the other side.

  9. SEPARABLE EQUATIONS Equation 2 • Then, we integrate both sides of the equation:

  10. SEPARABLE EQUATIONS • Equation 2 defines y implicitly as a function of 10. • In some cases, we may be able to solve for yin terms of ten.

  11. SEPARABLE EQUATIONS • We utilise the Chain Rule to justify this procedure. • If h and g satisfy Equation 2, and so

  12. SEPARABLE EQUATIONS • Thus, • This gives: • Thus, Equation 1 is satisfied.

  13. SEPARABLE EQUATIONS Example 1 • Solve the differential equation • Find the solution of this equation that satisfies the initial status y(0) = two.

  14. SEPARABLE EQUATIONS Example 1 a • We write the equation in terms of differentials and integrate both sides: • y2dy = x2dx • ∫y2dy = ∫x2dx • ⅓y3 = ⅓x3 + C • where C is an arbitrary constant.

  15. SEPARABLE EQUATIONS Case 1 a • Nosotros could have used a abiding C1 on the left side and some other constant C2 on the right side. • However, then, we could combine these constants by writing C = C2 – C1.

  16. SEPARABLE EQUATIONS Example 1 a • Solving for y, we get: • We could leave the solution like this or we could write information technology in the form where G = 3C. • Since C is an arbitrary abiding, and then is Chiliad.

  17. SEPARABLE EQUATIONS Example 1 b • If we put x = 0 in the full general solution in (a), we go: • To satisfy the initial status y(0) = two, we must take , and and so K = viii. • So, the solution of the initial-value trouble is:

  18. SEPARABLE EQUATIONS • The effigy shows graphs of several members • of the family of solutions of the differential • equation in Instance 1. • The solution of the initial-value trouble in (b) is shown in crimson.

  19. SEPARABLE EQUATIONS Example ii • Solve the differential equation

  20. SEPARABLE EQUATIONS East. yard. ii—Equation 3 • Writing the equation in differential class and integrating both sides, we have: (2y + cos y) dy = 6x2dx∫ (2y + cos y) dy = ∫ 6x2dx y2 + sin y = 2x3 + C • where C is a constant.

  21. SEPARABLE EQUATIONS Instance 2 • Equation three gives the general solution implicitly. • In this case, it's impossible to solve the equation to express y explicitly as a function of x.

  22. SEPARABLE EQUATIONS • The figure shows the graphs of several members of the family of solutions of the differential equation in Example 2. • Equally we expect at the curves from left to right, the values of C are:3, 2, 1, 0, -1, -2, -three

  23. SEPARABLE EQUATIONS Instance 3 • Solve the equation • y' = x2y • Kickoff, we rewrite the equation using Leibniz notation:

  24. SEPARABLE EQUATIONS Case 3 • If y≠ 0, we tin can rewrite it in differential note and integrate:

  25. SEPARABLE EQUATIONS Example 3 • The equation defines y implicitly as a part of x. • Even so, in this case, we tin solve explicitly for y. • Hence,

  26. SEPARABLE EQUATIONS Instance 3 • We can easily verify that the function y = 0 is also a solution of the given differential equation. • And so, we tin write the general solution in the form where A is an arbitrary constant (A = eC, or A = –eC, or A = 0).

  27. SEPARABLE EQUATIONS • The figure shows a direction field for the differential equation in Example 3. • Compare it with the next figure, in which nosotros use the equation to graph solutions for several values of A.

  28. SEPARABLE EQUATIONS • If you use the direction field to sketch solution curves with y-intercepts 5, two, i, –1, and –2, they will resemble the curves in the figure.

  29. SEPARABLE EQUATIONS Example four • In Section 9.2, we modeled the current I(t) in this electric circuit past the differential equation

  30. SEPARABLE EQUATIONS Example 4 • Find an expression for the electric current in a circuit where: • The resistance is 12 Ω. • The inductance is 4 H. • A battery gives a constant voltage of 60 Five. • The switch is turned on when t = 0. • What is the limiting value of the electric current?

  31. SEPARABLE EQUATIONS Example iv • With L = 4, R = 12 and E(t) = 60, • The equation becomes: • The initial-value problem is:

  32. SEPARABLE EQUATIONS Instance 4 • We recognize this as being separable. • We solve it as follows:

  33. SEPARABLE EQUATIONS Example 4 • Since I(0) = 0, nosotros have: 5 – ⅓A = 0 • So, A = xv and the solution is: I(t) = 5 – 5e-3t

  34. SEPARABLE EQUATIONS Example iv • The limiting current, in amperes, is:

  35. SEPARABLE EQUATIONS • The effigy shows how the solution in Case 4 (the current) approaches its limiting value.

  36. SEPARABLE EQUATIONS • Comparison with the other figure (from Department ix.2) shows that we were able to draw a fairly accurate solution bend from the direction field.

  37. ORTHOGONAL TRAJECTORY • An orthogonal trajectoryof a family of curves is a curve that intersects each curve of the family orthogonally—that is, at right angles.

  38. ORTHOGONAL TRAJECTORIES • Each member of the family y = mx of straight lines through the origin is an orthogonal trajectory of the family x2 + y2 = r2 of concentric circles withcenter the origin. • We say that the ii families are orthogonal trajectories of each other.

  39. ORTHOGONAL TRAJECTORIES Case 5 • Find the orthogonal trajectories of the family of curves x = ky2, where k is an capricious constant.

  40. ORTHOGONAL TRAJECTORIES Example 5 • The curves 10 = ky2 class a family of parabolas whose centrality of symmetry is the x-axis. • The first step is to discover a single differential equation that is satisfied by all members of the family.

  41. ORTHOGONAL TRAJECTORIES Example 5 • If we differentiate ten = ky2,we get: • This differential equation depends on k. • However,nosotros need an equation that is valid for all values of 1000 simultaneously.

  42. ORTHOGONAL TRAJECTORIES Instance 5 • To eliminate g, we annotation that: • From the equation of the given full general parabola x = ky2, we have thou = x/y2.

  43. ORTHOGONAL TRAJECTORIES Case 5 • Hence, the differential equation tin can be written every bit: or • This means that the gradient of the tangent line at any bespeak (x, y) on 1 of the parabolas is: y' = y/(2x)

  44. ORTHOGONAL TRAJECTORIES • On an orthogonal trajectory, the gradient of the tangent line must be the negative reciprocal of this gradient. • So, the orthogonal trajectories must satisfy the differential equation

  45. ORTHOGONAL TRAJECTORIES E. g. 5—Equation iv • The differential equation is separable. • Nosotros solve it as follows: • where C is an arbitrary positive abiding.

  46. ORTHOGONAL TRAJECTORIES Example 5 • Thus, the orthogonal trajectories are the family of ellipses given past Equation four and sketched here.

  47. ORTHOGONAL TRAJECTORIES IN PHYSICS • Orthogonal trajectories occur in various branches of physics. • In an electrostatic field, the lines of force are orthogonal to the lines of constant potential. • The streamlines in aerodynamics are orthogonal trajectories of the velocity-equipotential curves.

  48. MIXING Issues • A typical mixing problem involves a tank of stock-still capacity filled with a thoroughly mixed solution of some substance, such as salt. • A solution of a given concentration enters the tank at a fixed rate. • The mixture, thoroughly stirred, leaves at a fixed charge per unit, which may differ from the entering charge per unit.

  49. MIXING PROBLEMS • If y(t) denotes the amount of substance in the tank at time t, then y'(t) is the rate at which the substance is being added minus the charge per unit at which it is beingness removed. • The mathematical clarification of this situation oft leads to a offset-order separable differential equation.

  50. MIXING Bug • Nosotros can use the same type of reasoning to model a variety of phenomena: • Chemic reactions • Discharge of pollutants into a lake • Injection of a drug into the bloodstream

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