Only One Word for Review

1. Only One Word for Review Review Engineering Differential Equations The Second Test

2. Euler the Master of Us All

3. Euler’s Method: Tangent Line Approximation • For the initial value problem we begin by approximating solution y = (t) at initial point t0. • The solution passes through initial point (t0, y0) with slope f(t0,y0). The line tangent to solution at initial point is thus • The tangent line is a good approximation to solution curve on an interval short enough. • Thus if t1 is close enough to t0, we can approximate (t1) by

4. Euler’s Formula • For a point t2 close to t1, we approximate (t2) using the line passing through (t1, y1) with slope f(t1,y1): • Thus we create a sequence yn of approximations to (tn): where fn = f(tn,yn). • For a uniform step size h = tn– tn-1, Euler’s formula becomes

5. Euler Approximation • To graph an Euler approximation, we plot the points (t0, y0), (t1, y1),…, (tn, yn), and then connect these points with line segments. Note (t0, y0) was the IC

6. Autonomous Equations and Population Dynamics • In this section we examine equations of the form y' = f (y), called autonomous equations, where the independent variable t does not appear explicitly. • y’(t) is the gorillaz velocity which depends on gorillaz height y(t). Important heights are the rest points where y’ is zero; when f(y)=0. These are Equilibrium solutions. • Example (Exponential Growth): • Solution:

7. Autonomous Equations: Equilibrium Solns • Equilibrium solutions of a general first order autonomous equation y' = f (y) are found by locating roots of f (y) = 0. • These roots of f (y) are called critical points. • For example, the critical points of the logistic equation • are y = 0 and y = K. • Thus critical points are constant functions (equilibrium solutions) in this setting.

8. Autonomous Equations: Equilibrium Solns • Equilibrium solutions of a general first order autonomous equation y' = f (y) can be found by locating roots of f (y) = 0. • These roots of f (y) are called critical points. • Phase diagram is the y axis showing where the monkey climbs (f>0), rests (f=0) and falls (f<0) y’ = y(10 – y) • Thus critical points are constant functions (equilibrium solutions) in this setting.

9. Population Models • P(t)= fish pop size, b(t) = individ birth rate (births/unit time/fish) d(t) = individ death rate( deaths/unit time/fish) • Units for P’(t) are fish/unit time • Balance Law

10. Velocity and Acceleration • x(t) height of an object falling in the atmosphere near sea level; time t, velocity v(t) = x’(t), a(t) = x’’(t) accel. • Newton’s 2nd Law: Net F = ma = m(dv/dt) net force • Force of gravity: -mgdownward force • Force of air resistance: -  v (opp to v)upward force • Then get eqn for v (F = Force Grav + Resist Force) and x

11. Velocity and Acceleration • We can also get one eqn for x (using F = Force Grav + Resist Force) • m x’’ = -mg – ϒx’ is one second order de for x which is the same as the previous two first order DEs for x and v

12. Homogeneous Equations, Initial Values • Once a solution to a homogeneous equation is found, then it is possible to solve the corresponding nonhomogeneous equation. • Thus consider homogeneous linear Diff equations; and in particular, those with constant coefficients (like the previous) • Initial conditions typically take the form • Thus solution passes through (t0, y0), and slope of solution at (t0, y0) is equal to y0'.

13. Characteristic Equation • To solve the 2nd order equation with constant coefficients, we begin by assuming a solution of the form y = ert. • Substituting this into the differential equation, we obtain • Simplifying, and hence • This last equation is called the characteristic equation of the differential equation. • We then solve for r by factoring or using quadratic formula.

14. General Solution • Using the quadratic formula on the characteristic equation we obtain two solutions, r1 and r2. • There are three possible results: • The roots r1, r2 are real and r1r2. • The roots r1, r2 are real and r1 = r2. • The roots r1, r2 are complex. • First assume r1, r2 are real and r1r2. • In this case, the general solution has the form

15. Theorem • Suppose y1 and y2 are solutions to the equation and that the Wronskian is not zero at the point t0 where the initial conditions are assigned. Then there is a choice of constants c1,c2 for which y = c1y1 + c2 y2 is a solution to the differential equation (1) and initial conditions (2).

16. Linear Independence and the Wronskian • Two functions f and g are linearly dependent if there exist constants c1 and c2, not both zero, such that for all t in I. Note that this reduces to determining whether f and g are multiples of each other. • If the only solution to this equation is c1 = c2 = 0, then f and g are linearly independent.

17. Theorem • If f and g are differentiable functions on an open interval I and if W(f, g)(t0)  0 for some point t0 in I, then f and g are linearly independent on I. Moreover, if f and g are linearly dependent on I, then W(f, g)(t) = 0 for all t in I.

18. Repeated Roots • Recall our 2nd order linear homogeneous ODE • where a, b and c are constants. • Assuming an exponential soln leads to characteristic equation: • Quadratic formula (or factoring) yields two solutions, r1 & r2: • When b2 – 4ac = 0, r1 = r2 = -b/2a, since method only gives one solution:

19. General Solution When roots are the same, then two linearly independent solutions are e^{rt} and te^{rt} • Thus the general solution for repeated roots is

20. Complex Roots of Characteristic Equation • Recall our discussion of the equation where a, b and c are constants. • Assuming an exponential soln leads to characteristic equation: • Quadratic formula (or factoring) yields two solutions, r1 & r2: • If b2 – 4ac < 0, then complex roots: r1 =  + i, r2 =  - i • Thus

21. Euler’s Formula; Complex Valued Solutions • Substituting it into Taylor series for et, we obtain Euler’s formula: • Generalizing Euler’s formula, we obtain • Then • Therefore

22. Real Valued Solutions: The Wronskian • Thus we have the following real-valued functions: • Checking the Wronskian, we obtain • Thus y3 and y4 form a fundamental solution set for our ODE, and the general solution can be expressed as

23. Real Valued Solutions • Our two solutions thus far are complex-valued functions: • We would prefer to have real-valued solutions, since our differential equation has real coefficients. • To achieve this, recall that linear combinations of solutions are themselves solutions: • Ignoring constants, we obtain the two solutions

24. Theorem (Nonhomogenous Des) • If Y1, Y2 are solutions of nonhomogeneous equation then Y1 - Y2 is a solution of the homogeneous equation • If y1, y2 form a fundamental solution set of homogeneous equation, then there exists constants c1, c2 such that

25. Theorem (General Solution) • The general solution of nonhomogeneous equation can be written in the form where y1, y2 form a fundamental solution set of homogeneous equation, c1, c2 are arbitrary constants and Y is a specific solution to the nonhomogeneous equation.

26. Thanks • Leonhard Euler for his insights and beautiful mathematics. You are number –e^{(pi)i} in my book. • Ry Cooder’s music I think it’s going to work (as long as you do) • Good luck on Test 2 42