1. Math for CS Second Order Linear Differential Equations
Lecture 8
Second Order Linear Differential Equations
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

2. Second Order Differential Equations
Math for CS
Second Order Differential Equations
A second order differential equation is an equation involving the unknown function y, its derivatives y' and y'', and the variable x:
We will only consider explicit differential equations of the form,
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

3. Arbitrary constants of Second Order Equations
Math for CS
Arbitrary constants of Second Order Equations
Since a second order equation involves a second derivative and hence, two integrations are required to find a solution, the solution usually contains two arbitrary constants.
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

4. Equations with missing
Math for CS
Equations with missing
Nonlinear second order differential equations are difficult to solve for general f:
but there are two cases where simplifications are possible:
Equations with the y missing
Let v = y'. Then the new equation satisfied by v is
This is a first order differential equation. Once v is found its integration gives the function y.
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

5. Equations with missing
Math for CS
Equations with missing
(2) Equations with the x missing
Let v = y'. Since
We get
This is again a first order differential equation. Once v is found then we can get y through
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

6. Math for CS
The Linear Case
A linear second order differential equations is written as
associate the so called associated homogeneous equation
For the study of these equations we consider the explicit ones given by
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

7. Solution of the Linear Case
Math for CS
Solution of the Linear Case
where p(x) = b(x)/a(x), q(x) = c(x)/a(x) and g(x) = d(x)/a(x). If p(x), q(x) and g(x) are defined and continuous on the interval I, then the IVP
has a unique solution defined on I.
The general solution to the equation (NH) is given by
where is the general solution to the homogeneous associated equation (H); is a particular solution to the equation (NH).
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

8. Solution of the Linear Case 2
Math for CS
Solution of the Linear Case 2
In conclusion, we deduce that in order to solve the nonhomogeneous equation (NH), we need to :
Step 1: find the general solution to the homogeneous associated equation (H), say
Step 2: find a particular solution to the equation (NH), say
Step 3: write down the general solution to (NH) as
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

9. Homogeneous Linear Equations
Math for CS
Homogeneous Linear Equations
Consider the homogeneous second order linear equation
or the explicit one
Basic property:If and are two solutions, then
is also a solution for any arbitrary constants and
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

10. Linear Independence of Solutions
Math for CS
Linear Independence of Solutions
Let y1 and y2 be two differentiable functions. The Wronskian W(y1,y2), associated to y1 and y2, is the function
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

11. Properties of the Wronskian
Math for CS
Properties of the Wronskian
(1) If y1 and y2 are two solutions of the equation y'' + p(x)y' + q(x)y = 0, then
(2) If y1 and y2 are two solutions of the equation y'' + p(x)y' + q(x)y = 0, then
In this case, we say that y1 and y2 are linearly independent.
(3) If y1 and y2 are two linearly independent solutions of the equation y'' + p(x)y' + q(x)y = 0, then any solution y is given by y=c1y1+c2y2
for some constant c1 and c2. In this case, the set {y1,y2}is called the fundamental set of solutions.
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

12. Homogeneous Linear Equations with Constant Coefficients
Math for CS
Homogeneous Linear Equations with Constant Coefficients
A second order homogeneous equation with constant coefficients is written as
where a, b and c are constant. This type of equation is very useful in many applied problems (physics, electrical engineering, etc..). Let us summarize the steps to follow in order to find the general solution:
(1) Substitute Then we can write down the characteristic equation
This is a quadratic equation. Let r1 and r2 be its roots.
If r1 and r2 are distinct real numbers (if D>0), then the general solution is
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8

13. Equations with Constant Coefficients 2
Math for CS
Equations with Constant Coefficients 2
(3) If r1=r2 (which happens if D=0 ), then the general solution is
(4) If r1 and r2 are complex numbers (which happens if D<0), then the general solution is
where
Outline:
Central Scientific Problem – Artificial Intelligence
Machine Learning:
Definition
Specifics
Requirements
Existing Solutions and their limitations
Multiresolution Approximation: Limitation
Our Approach. Results. Binarization. Plans.
Math for CS
Lecture 8