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

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

.

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

.

Figure 1: Graph of

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

*Example 2*: Alice throws a ball straight up at a speed of *u* m/s. Assume that the acceleration due to gravity equals -10m/s^{2}. 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/s^{2} *means*), we can write the differential equation as . Solving this, and using the fact that the initial velocity is *u* gives us that .

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

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 V_{1} across the resistor equals the current I times the resistance R, or, V_{1}=IR

The equation for a capacity implies that the voltage V_{2} across the capacity equals charge q divided by the capacitance C, or, V_{2}=q/C

Finally, because of Kirchoff's voltage law*, we have that E, the voltage of the battery, equals V_{1}+V_{2}.

Thus . 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

(A)

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

* 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) is a 1st order ODE of degree 1.

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

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

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

**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: .

**Example 3.**

(a) is a Linear ODE.

(b) is a NON-Linear ODE since the coeff. of *y* is *y*.

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

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

**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) is a general solution of .

(b) is a general solution of .

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) is a particular solution of .

(b) is a particular solution of .

Exercise 4. Are the solutions below general or particular?

**5 Solving ODEs by Geometric Methods**

By graphing the *direction fields*

E.g. (http://math.rice.edu/ dfield/dfpp.html)

Figure 2: Graph of

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 and by using MAXIMAs plotdf command.

**6 Separable Differential Equations**

** ** or

** **

E.g. , ,

Then we can solve it by integrating it directly

Example 6. Consider

Therefore, the solution is

Exercise 6. Solve

**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

Let ,

Substitute into (1) becomes

Therefore,

Exercise 7. Solve

**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:

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

And, *y*_{0},y_{1}(x),y_{2}(x)......,*y*_{n}(x) represents a sequence of approximations for *y(x)*.

Example 11. To find the approximate solutions to , , we have: , and

, Substituting this into:

we have

Starting from , we have:

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 , when

**10 Exact Differential Equations (Advanced Topic)**

A first order ODE

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

of some function u(x, y).

Therefore,

Since ,

is the solution.

**10.1 Check for exactness**

Check for exactness:

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) is not exact, since and

(b) is an exact differential equation, since

Let and ,

and

Find the solution ,

Compare to ,

Therefore, the solution is

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

* *

* *

**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:

*F* is the Integrating factor.

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

* *

Let and ,

and

And the solution is

Exercise 10.2. Solve

**11 Existence and Uniqueness (Advanced Topic)**

Consider the differential equation with initial condition

**11.1 Existence Theorem**

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

and* f(x, y) *is bounded in *R*, i.e. |*f(x, y)*| ≦ K for all

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 .

**11.2 Uniqueness Theorem**

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

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 .

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,

take the rectangle . Then , and

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

**Example 11.2. No solutions**

A initial value problem has no solution, since

is not defined when x = 0.

**Example 11.3. Non-Uniqueness.**

Consider the initial value problem

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

However,

is undefined when x = 0.

Exercise 11. Prove that have Existences and Uniqueness solution in

(a) . 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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