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1

Second order linear homogeneous differential equations- general form

d2y/dt2 + P(x) (dy/dt) + Q(x)y =0

2

Second order linear homogeneous differential equations- ways of solving

•if constant coeffs auxiliary sin cos sinh cosh
****Boundary value problems: unknown coeffs consider cases for eigenfunctions

•the regular Sturm-Liouille problem

•self adjoint form:
****multiply by integrating factor

•normal/canonical form: without boundary cond
**** setting y=uv to show possible v and finding general solution

•reduction of order: given one solution or finding one by any other method

•power and Taylor series
•Frobenius series

3

Second order linear homogeneous differential equations- constant coefficients

Auxiliary equation: MAKE SURE
From y=e^(mx)
Form auxiliary then solve or use b^2 -4ac.

General solution with 2 constants:
1 repeated real solution: m
y=(A +Bx) e^(mx)

2 distinct real solutions:m,n
y=Ae^(mx) + Be^(nx)
Also: if m= plus or minus μ
y= Asinhμ + Bcoshμ

Two complex conjugate: a plus or minus bi
y= e^(a) [Asinbx + Bsinbx]

4

Second order linear homogeneous differential equations- boundary value problems

Boundary value problems are defined on interval [a,b] with conditions as a result of a and b e.g. Finding the value of λ in constant such that the ODE has Non trivial solutions given conditions.

CASES λ > 0 λ=0 and λ< 0
By setting λ =μ^2
Using boundary conditions after auxiliary to see if non trivial. Eigenfunctions found for a set of λ_1 < λ_2 < ...
found from intersections if of two graphs e.g.lhs and rhs of..
e.g. Tan μ = 1/μ e.g. Tan( √λ) = 1/√λ
To give eigenfunction in terms of λ as λ = μ^2

5

The regular Sturm-Lioville problem

Let a,b be real values with a< b.
Let p(x), q(x), r(x) be functions that are continuous on [a,b] and suppose p(x) > 0, r(x) > 0. Then the regular sturm-lioville problem is stated as:
FIND ALL VALUES OF THE CONSTANT λ FOR WHICH THE ODE HAS NON TRIVIAL SOLUTIONS SATISFYING THE BOUNDARY CONDITIONS

(py')' + (λr +q)y = 0

For constants:
A_1y(a) + A_2y'(a) =0
B_1y(b) + B_2y'(b) = 0

And tending to infinity as n does
For each there exists an eigenfunction
NOTÉ: No mixed boundary conditions and y' value from product rule only

We have values of λ forming a discrete set if they exist, each λ_1 < λ_2 < ...< λ_n

6

Orthogonalitu of eigenfunctions
In 2nd ODE

Theorem:

If y_m(x) and y_n(x) are eigenfunctions of SLBVP associated with eigenvalues λ_m and λ_n then:


(py_m')' + (λ_m r +q)y_m = 0
And
(py_n')' + (λ_n r +q)y_n = 0
Then THEOREM: The functions y_m(x) and y_n(x) are orthogonal. I.e. If m ≉ n then
Integral_from_a_to_b( r(x)y_m(x)y_n(x) .dx)


Proof: by the product rule multiplying and subtracting we can show multiple of (λn - λm) this is 0.

7

Self adjoint form definition: linear ode homogenous

(RsLP is written is self adjoint form)

Standard: y" + P(x)y' + Q(x)y = 0
Self-adjoint form:
By multiplying by
p(x) = exp(integral P(x).dx)

SELF ADJOINT FORM:
(py')' +sy =0

8

Normal form or canonical form

To solve a general homogenous linear second order ode: WITHOUT BOUNDARY CONDITIONS

y" + P(x)y' + Q(x)y =0

Setting y=u(x)v(x) to show possible to choose v such that it takes the form
(NORMAL CANONICAL):
u" + f(x)u =0

I.e. u' disappears
Note can divide

Note may solve this one by: IF or auxiliary

9

The method of reduction of order

Finding another solution given/after solving another

May have to find 2 solution:

y" + P(x)y' + Q(x)y =0
Suppose we have one solution y(x) = v(x)
u = integral_x( (c_2/v^2(t)) • exp( -integral_t( P(s).ds).dt +c

Note substitute and dummy variable

Constant c_2 c


If y(x) us linearly independent v(x) then we have another solution to the OdE

10

Power series

Power series about x= x_0



Value of the limit


Radius of convergence

11

Ratio test and radius of convergence

Ratio test:

12

Taylor series examples:

Note can substitute: