IZTECH Math255 Notes p4

By E. Z.

Forced Mechanical Vibrations

$\ F_ {net}=m\vec{u}''(t)=\vec{F}_{gravitational}+\vec{F}_{restoring}+\vec{F}_{damping}+\vec{F}_{external}$

$\ mu''+bu'+ku=f(t) $
$\ u(0)=u_0 $$\ u'(0)=u_1 $

Reduction of Order Method

Given one solution $\ y_1 $ to the homogenous equation: $\ y''+p(x)y'+q(x)y=0 $, there is needed to be find $\ y_2 $ such is linearly independent from $\ y_1 $.

$\ y_2=u(x)y_1(x), W(y_1, y_2 ) ≠ 0 $
Let us apply the transformation:
independent variable $x$ to $λ$
dependent variable $y$ to $u$
Then the DE becomes: $\ u'' y_1 + u' (2y'_1 + p(x) y_1)=0 $
By setting $w$ $=$ $u$ $'$ the DE becomes:
$\ w' + [\frac{2y'}{y}+p(x)]=0 $
which is a first order linear ODE & easy to solve.
After finding $w$ we can transform it back to $y$ .

Series Solution of Second Order DE's

If it is not possibe to find fundamental solutions for $y$ , using power series approximation to the position $\ \small x_0 $:

\[\ \lim_{m \to ∞}( \sum_{n=0}^{m} a_n (x-x_0)^n ), x ∈ 𝕀 \]

\[\ \frac{d}{dx} (\sum_{n=0}^{∞} (a_n(x-x_0)^n)) \] does not always equal to \[\ (\sum_{n=0}^{∞} \frac{d}{dx} (a_n(x-x_0)^n)) \]

If:
(

i

) $\ \sum_{k=1}^{∞} u_k(x_o) $ converges at some point $\ x_o ∈ (a,b) $
(

i

i

) $\ \sum_{k=1}^{∞} u'_k(x) $ converges uniformaly on $\ (a,b) $ to $\ f(x)= \sum_{k=1}^{∞} u'_k(x) $
Then:
(1) The series $\ \sum_{k=1}^{∞} u_k(x) $ converges at every $\ x ∈ (a,b) $ and the sum
$\ F(x)= \sum_{k=1}^{∞} (u_k (x)) $ is differantiable with $\ F'(x)=f(x) $ for every $\ x ∈ (a,b) $
(2) Convergence of $\ \sum_{k=1}^{∞} u_k(x) $ to $\ F(x) $ is uniform on $\ (a,b) $
SO:
\[\ \sum_{k=1}^{∞}u_k(x)=F(x) ⇒ \sum_{k=1}^{∞}u'_k(x)=f'(x) \]

Note: This theorem is not one of the achivements of this course but it is important to know to apply the method.

Series Solution near an ordinary point

\[\ A(x)y''+B(x)y'+C(x)y=0 \]

A point $\ x_0 $ with the property $\ A(x) ≠ 0 $ is called an ordinary point, otherwise it is called to be singular point.

Notice for ordinary points DE can be written as: \[\ y''+p(x)y'+q(x)y=0 \]

Note: As far as this cource concerns we will only be solving series solutions for ordinary points.

Suppose \[\ y(x)= \sum_{k=0}^{∞}c_k(x-x_0)^k \] By supposing it satisfies the necessary conditions: \[\ y'(x)= \sum_{k=0}^{∞}c_kk(x-x_0)^{k-1} \text{ and } y''(x)= \sum_{k=0}^{∞}c_kk(k-1)(x-x_0)^{k-2} \] By substituting these formulas in place at DE, we can find $\ c_k $