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MIT 18.03 Differential Equations with Arthur Mattuck

PDF Link	Lecture Topics
Lecture 1	First order ODEs of the form y' = f(x,y)
Lecture 2	Numerical solutions to differential equations by Euler's Method
Lecture 3	First-order linear equations and the separation of variables
Lecture 4	Change of variables and substitutions
Lecture 5	Autonomous differential equations, integral curves, critical points
Lecture 6	Complex numbers, polar form
Lecture 7	Linear first order ODEs
Lecture 8	Linear ODEs: Mixing, RC Circuits, Growth and Decay Models
Lecture 9	xxx
Lecture 10	xxx
Lecture 11	xxx
Lecture 12	xxx

Lectures

Lecture 1:

Topics: first order ODEs of the form y' = f(x,y),
Geometric view of differential equation (y' is direction field, solution y is integral curve)
To draw direction field, choose constant and solve f(x,y)=c
- [all solutions have same slope along this curve -- this is called an isocline]
Example: y' = -x/y with direction field, integral curves, and analytical solution
Example y' = 1+x-y with isoclines and integral curves
Existence and uniqueness theorem introduced
Example: dy/dx = y-1/x, existence not guaranteed at x=0

Lecture 2:

Topics: numerical solutions to differential equations by Euler's Method
Euler's Method: begin at initial point (x0) with initial slope (A0). Increment by step size (h) and calculate new slope (An) at this new point (xn, yn). Continue method until stable.

We get: y_n+1 - y_n = h * A_n,
So our Euler Equations are:

x_n+1 = x_n + h
y_n+1 = y_n + h*A_n
A_n = f(x_n, y_n)

How good is Euler's Method?

If solution is a straight line, the Euler's Method will produce exact solution.
If solution is convex, Euler's method will produce results that are too low.
If solution curve is concave, Euler's method will estimate solutions that are too high.
We can address this problem without plotting the solution by taking the second derivative of the solution (or simply using the given differential equation and initial value).
We know a lot about the form of the solution simply by looking at the differential equation without analytically calculating a solution.

How do we improve accuracy of numerical estimate of solution?

Halving the error halves the error

Use a better estimate of the slope

Runge-Kutta 2nd Order Method (RK2, RK4)
(also called Modified Euler's Method)

Lecture 3:

Topics: first-order linear equations and the separation of variables
Linear equations take the form:

a(x)y' + b(x)y = c

The standard linear form is:

y' + p(x)y = q

Equations in standard linear form (SLF) can always be solved. This form makes the physical interpretation meaningful as sources are on the right-hand side.
Examples:

Conduction cooler problem: dT/dt = k(Te - T)
Diffusion model: dC/dt = k(Ce-C)

Solving DE in SLF

1. Put equation in standard form, y' + p(x)y = q,
2. Find integrating factor u(x) such that

u = exp( int(p dx) ) is our integrating factor
we want u' = pu so we can use derivative rules
uy' + puy = qu,

3. Multiply both sides of equation by u(x),
4. Integrate and solve.

Solutions to DE composed of steady-state solution and transient

Lecture 4:

Topics: change of variables and substitutions
Change of variable #1: scaling:

For example: x1 = x/a, y1 = y/b
we scale to: change units, make variables dimensionless, simplify constants

Direct substitutions are those where the new variable combine old variables.
Indirect substitutions are those values where old variables are a combination of new variables.
Bernoulli Equations are those ODEs of the form:

y' = p(x)*y + q(x)*y^n
Solve by dividing by y^n, this will produce a linear equation upon substitution

Let v = 1/(y^n-1),
then v' = (1-n) * y' / y^n
and so we have the linear equation: v' / (1-n) = p(x)*v + q(x)
Solve and convert back to y from v

Homogeneous 1st Order ODEs are those in the form:
- y' = F(y/x)
- For example we can rewrite y' = x^2*y / (x^3 +y^3) as y' = x/y / (1 + (x/y)^3)

Lecture 5:

Topics: autonomous differential equations, integral curves, critical points
Autonomous Differential Equations:

Right-hand side does not contain independent variable
Difficult to integrate directly
Integral curves are invariant under translation (easy to plot)

Problem: want to get qualitative information about system without solving
Plot Direction Field

1. Find critical points: solve f(y0) = 0
2. Graph f(y) to see slope

Logistic Growth Equation:
- dy/dt = ky given k = a-by
- f'(y) = ay - by^2

Lecture 6:

Topics: complex numbers, polar form
Euler's Relation:

exp(i*theta) = cos(theta) + i*sin(theta)

Exponents satisfy:

Exponential Law:

a^x * a^y = a^(x+y)

Only function we have that has this property:

For a solution y = exp(at), dy/dt = ay
That is, it's derivative is itself again (times constant term)

Infinite series should behave nicely

Polar form is good for multiplication of complex numbers

Lecture 7:

Topics: linear first order ODEs
Special Form:

y' + ky = k q_e(t)

Solutions to 1st order differential equations

Sum of steady-state and transient solutions

1st order ODE with trigonometric Input

Problem:

y' + ky = k * cos(omega * t)

Easiest to solve in the complex domain (integration is easier)
Use the real part of Euler's formula

Now, y' + ky = k e^(j * omega * t)
Solution is y = y1 + y2

Final solution:
- y = cos(omega*t - phi) / sqrt(1+ (omega/k)^2)
- phi = arctan(omega / k) [phase lag]

Lecture 8:

Topics: Linear ODEs: Mixing, RC Circuits, Growth and Decay Models
Trig Identity: a*cos(theta) + b*sin(theta) = C*cos(theta-phi)
Basic Linear ODEs

1. y' + k*y = k*q_e(t) [Conduction-Diffusion Model]
2. y' + ky = q(t)
3. y' + k(q)*y = q(t) [k not constant, solve by integrating]
k > 0

Mixing Example: Flow of Salt in Water Bath

x: amount of salt in bath, Ce: concentration of incoming salt, r: flow rate, v: volume of tank
rate of change of salt in bath = inflow - outflow
dx/dt = r*Ce - r*x/v [intuitive form]
dx/dt + r*x/v = r*Ce [standard form]
let Ct = x/v [concentration of salt]
V*dCt/dt + r*Ct = r*Ce
dCt/Dt + r/v*Ct = r/v*Ce [now in form 1 if k=r/v - solvable!]

Linear Equation 2: RC Circuit

q: charge, i = dq/dt
rate of change of salt in bath = inflow - outflow
R*dq/dt + q/c = V(t)
q' + q/RC = V/R [standard form]

Linear Equation 3: Population Growth

dP/dt = aP
P' - aP

Lecture 9:

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Lecture 10:

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Lecture 11:

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Lecture 12:

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