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Chapter 1 First order differential equations

Page history last edited by sidjaggi 13 years, 5 months ago

Chapter 1 First order differential equations

 

1 Differential Equations

 

Example 1: Alice runs toward Bob at 10m/s. She is initially 100m away from him. How long will it take her to get there? What does the graph of her motion look like?

Solution via differential equations: Let time be denoted by t, Alice's position as a function of time be x(t), the time required for Alice to teach Bob be T. Then, since Alice's velocity is constant and equals 10, then

                                                                               Formula

Then, writing the differential equation in separable form and integrating both sides we get

                                                                           Formula.

When t = 0, x(0) = 0 Formula c = 0. Therefore,

                                                                              Formula.

Figure 1: Graph of Formula

 

When x(t) = 100, Alice meet Bob. So

                                                                                Formula

 

 

Example 2Alice throws a ball straight up at a speed of u m/s. Assume that the acceleration due to gravity equals -10m/s2. Plot the position of the ball as a function of time.

Solution: The first differential equation to write involves the velocity of Alice's ball. Since we know its velocity decreases 10m/s every second (that's what acceleration of -10m/s2 means), we can write the differential equation as Formula. Solving this, and using the fact that the initial velocity is u gives us that Formula.

 

Next, we note that velocity itself is the derivative of the y-coordinate with respect to time. Thus we can writeFormula. Solving this, and using the fact that the height at time 0 equals 0, gives us that Formula.

 

Here's a plot of this function. Also some MAXIMA code to plot it as a function of u.

load(draw);

with_slider_draw( /* first two arguments are the parameter and parameter values */ u,makelist(i,i,10,200)/10, /* the rest of arguments describe the function */ explicit(u*t-5*t^2, t, 0, u/5), yrange = [0, 22], xrange = [0, 5] );

 

Note 1: To see how the curve changes as a function of u, after MAXIMA plots it, click on the image, and use the slider on the top of the MAXIMA window to change the value of u.

 

Note 2: For more innovative uses of MAXIMA, check out this link below.

http://wxmaxima.sourceforge.net/wiki/index.php/Animations

 

 

Example 3: Consider an RC-circuit (Resistor-Capacitor circuit).

 

The equation for a resistor implies that the voltage V1 across the resistor equals the current I times the resistance R, or, V1=IR

The equation for a capacity implies that the voltage V2 across the capacity equals charge q divided by the capacitance C, or, V2=q/C

Finally, because of Kirchoff's voltage law*, we have that E, the voltage of the battery, equals V1+V2.

Thus Formula. Differentiating this equation with respect to time t and noting that dq/dt = I (since current equals the change of charge as a function of time) gives us

Formula        (A)

Writing this in a separable form and solving gives us Formula for some constant A. But noting that at time t=0, the current equals E/R, we find that Formula. Hence Formula.

 

http://en.wikipedia.org/wiki/Kirchhoff's_circuit_laws

 


Exercise: As in Example 2, try using MAXIMA to plot the solution of the differential equation (A) in Example 3 for fixed values of E and C, and varying R.

 

Exercise: Try using MAXIMA to solve the differential equation (A) itself.

 

 

 

 

 

 

 

2 Order and Degree


2.1 Order
The order of a differential equation is the "order" of the highest derivative that appears in the equation.


2.2 Degree
The degree of a differential equation is the power of the highest derivative term.


Example 2.

 

(a) Formula is a 1st order ODE of degree 1.

(b) Formula is a 2nd order ODE of degree 1.

(c) Formula is a 3rd order ODE of degree 2.

 

Exercise 2. What are the orders and degrees of the ODEs below?

  1. Formula
  2. Formula
  3. Formula
  4. Formula

 

 

 

 

 

 

3 Linearity
A Differential Equation is called linear if there are no multiplications amongst dependent variables and/or their derivatives. In other words, all coefficients are functions of independent variables.

General form: Formula.

 

Example 3.

 

(a) Formula is a Linear ODE.
(b) Formula is a NON-Linear ODE since the coeff. of y is y.
(c) Formula is a NON-Linear ODE since the coeff. of y'' is y'.

 

Exercise 3. Are the below ODEs Linear or Non-Linear?

 

  1. Formula
  2. Formula
  3. Formula
  4. Formula

 

 

 

 

 

 

 

4 General and Particular Solutions


4.1 General Solution


Solutions obtained from integrating the dierential equations are called general solutions. The general solution of an n-th order ordinary dierential equation contains n arbitrary constants resulting by integrating n times.


Example 4.1.

 

(a) Formulais a general solution of Formula.
(b) Formula is a general solution of Formula.


The general solution represents a family of curves, each curve corresponds to the setting of the constant(s) to a specific value.

 

4.2 Particular Solution


Particular solutions are the solutions obtained by assigning specic values to the arbitrary constants in the general solution.


Example 4.2.


(a) Formula is a particular solution of Formula.
(b) Formula is a particular solution of Formula.

 

 

Exercise 4. Are the solutions below general or particular?

 

  1. Formula
  2. Formula
  3. Formula
  4. Formula

 

 

 

 

 

 

 

5 Solving ODEs by Geometric Methods


By graphing the direction fields
E.g. Formula(http://math.rice.edu/ dfield/dfpp.html)

Figure 2: Graph of Formula

The MAXIMA code for this is plotdf(2*x) 

Try clicking on the graph after the direction field is plotted, to find particular trajectories.

Many other options can be found at this link http://maxima.sourceforge.net/docs/manual/en/maxima_68.html

 

Exercise 5. Solve Formula and Formula by using MAXIMAs plotdf command.

 

 

 

 

 

 

6 Separable Differential Equations

 

                                           Formula  or  Formula

 

E.g. Formula, Formula, Formula
Then we can solve it by integrating it directly


Example 6. Consider Formula

                                                                      Formula

FormulaTherefore, the solution is

                                                                              Formula

 

Exercise 6. Solve Formula

 

 

 

 

 

 

7 Reduction to separable form by Substitution
Certain nonseparable ODEs can be made separable by transformations that introduce for y a new unknown function.


Example 7. Solve

                                                          Formula

Let Formula, Formula

Substitute into (1) becomes

                                                                           Formula

Therefore,

                                                                                Formula

 

Exercise 7. Solve Formula

 

 

 

 

 

 

8 Modeling Problem


Exercise 8. Consider the amount of Carbon-14 in a living tree is 0.4 g. By the radiocarbon dating, we find that a fossilized tree contain 0.02 g Carbon-14. How old is the fossilized tree? (Carbon-14 has a half-life of 5730 years.)

 

 

 

 

 

 

 

9 Picard Iterations
Consider the differential equation with an initial condition:

                                                            Formula

In general, at the n-th iteration, we have:

                                                                 Formula

And, y0,y1(x),y2(x)......,yn(x) represents a sequence of approximations for y(x).


Example 11. To find the approximate solutions to FormulaFormula, we have: Formula, and
Formula, Substituting this into:

                                                                 Formula

we have

                                             Formula

Starting from Formula, we have:

          Formula

 

Here's some MAXIMA code to do this 

 

series: x$

for p: 1 unless p > 2 do

(series: integrate(1+(series)^2,x))$

series;

 

Play around with this. See if you can write a slider-based MAXIMA function to compute the n-th Picard iteration, and plot it.

 

Exercise 11. Use Picard's method to solve FormulaFormula when Formula

 

 

 

 

 

 

 

10 Exact Differential Equations (Advanced Topic)
A first order ODE

                                                                 Formula

is called an exact differential equation if differential form Formula is exact, i.e. this form is the differential

Formulaof some function u(x, y).
Therefore,

                                                                 Formula

Since Formula,

                                                                                          Formula

is the solution.


10.1 Check for exactness
Check for exactness:

                                                                                     Formula

if equal, it is exact. We can solve it by find u(x, y).
Otherwise, it is not exact and we need to find the integrating factor.


Example 10.1.


(a) Formula is not exact, since Formula and Formula
(b) Formula is an exact differential equation, since
Let Formula and Formula,

                                                                 Formula  and  Formula
Find the solution Formula,

                                                    Formula

                                                            Formula

Compare to Formula,

                                                                                     Formula

Therefore, the solution is

                                                                                Formula

 

Exercise 10.1. Is the following differential equation exact? If so, solve it by finding u.

 

                                                            Formula

 

 

 

 

 

 

 

10.2 Integrating Factor
Fortunately, we have methods to find solutions for such equations, but only for integrating factors depending only on one variable,
i.e., F(x) or F(y).
The formulae for the integrating factors are shown below:

                              Formula

F is the Integrating factor.

 

Example 10.2. Recall the above equation, Formula. If we multiply it by Formula, we get an exact equation.

 

Let Formula and Formula ,

                                                           Formula  and  Formula

And the solution is

                                                                           Formula

 

Exercise 10.2. Solve Formula

 

 

 

 

 

 

 

 

11 Existence and Uniqueness (Advanced Topic)

Consider the differential equation with initial condition

     Formula

 

11.1 Existence Theorem
If f(x, y) is continuous at all points (x, y) in some rectangle R defined as

                                                            Formula

and f(x, y) is bounded in R, i.e. |f(x, y)| ≦ K for all Formula
then, the initial problem (2) has AT LEAST ONE solution y(x) and this solution is defined at least for all x in the interval Formula.

 

11.2 Uniqueness Theorem
Let f(x, y) and its partial derivative Formula be continuous for all (x, y) in the rectangle R and bounded, say,

                         Formula

then, the initial problem (2) has AT MOST ONE solution y(x). This together with the Existence Theorem implies there is exactly ONE solution for y(x).
Again, this solution is defined at least for all x in the interval Formula.

Figure 3: The codition of the existence and uniqueness theorem. (a) First case. (b) Second case

 

Example 11.1. Existness and Uniqueness.
Consider the initial value problem,

                                                                        Formula

take the rectangle Formula. Then Formula, and

                                                            Formula

 

So, the initial value problem has at most one solution y(x) and it exists at least for all x in that subinterval Formula.

 

Example 11.2. No solutions
A initial value problem Formula has no solution, since

                                                                                Formula

is not defined when x = 0.

 

 

Example 11.3. Non-Uniqueness.
Consider the initial value problem

                                                                           Formula

take the rectangle Formula. Then a = 2, b = 8, and

                                                                           Formula

However,

                                                                                Formula

is undefined when x = 0.


Exercise 11. Prove that Formula have Existences and Uniqueness solution in

(a) Formula. Then, solve it when y(0)=1.

 

 

 

 

 

 

(b) any other interval that chosen by yoursself.

 

 

 

 

 

 

(c) every interval.

 

 

Comments (1)

sidjaggi said

at 1:35 pm on Sep 20, 2010

Thanks to Andy of 2013, for finding an small typo in the Picard's iteration method notes.

Also, some of you pointed out that there are no practice problems for the step method. This is true. Try solving problem 2 in Problem Set 1, which is for the direction field method, also for the step method -- just two steps each.

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