2nd Order Linear Differentials

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Table of Contents

• 2nd Order Linear Differentials
  • Key Definitions
    • \(y_1(t)=\cos\omega t \quad \text{and} \quad y_2(t)=\sin\omega t\)
    • \(y(t)=C_1\cos\omega t \space + \space C_2\sin\omega t\)
    • \(C_1 y_1 + C_2 y_2:I\to\mathbb R\)
    • \(C_1 y_1 + C_2 y_2 = 0\)
    • \(y(t;C_1,C_2)=C_1 y_1 + C_2 y_2\)
  • Existence and Uniqueness Theorem: 2nd, Linear
  • Wronskian
    • \(W(t):= \det\begin{bmatrix} u(t) & v(t) \\ u'(t) & v'(t)\end{bmatrix}:=u(t)v'(t)-v(t)u'(t)\)

#UCLA #Y1Q3 #Math33B

2nd Order Linear Differentials

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Key Definitions

Second-Order Linear Differential Equations - diff. eq. of the form: \(y''(t)+p(t)y'+q(t)y=g(t)\) Where $p, q, g$ are coefficient functions and $g(t)$ is the forcing term

If $g(t)=0$, the diff. eq. is homogenous

E.g. Simple Harmonic Motion: \(y''+\omega^2y=0\)

\(y_1(t)=\cos\omega t \quad \text{and} \quad y_2(t)=\sin\omega t\)

\(y(t)=C_1\cos\omega t \space + \space C_2\sin\omega t\)

Linear Combination - lin. comb. of 2 func. $y_1,y_2$:

\(C_1 y_1 + C_2 y_2:I\to\mathbb R\)

Linearly Independent - $y_1,y_2: I\to\mathbb R$ are lin. indep. if:

\(C_1 y_1 + C_2 y_2 = 0\)

for all $t\in I$ else the funcs. are linearly dependent

Fundamental Set of Solutions - if $y_1,y_2$ are lin. indep. solutions to some 2nd order lin. diff. eq., and they “generate” all other sols., then the general solution is:

\(y(t;C_1,C_2)=C_1 y_1 + C_2 y_2\)

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Existence and Uniqueness Theorem: 2nd, Linear

Sps. $p,q,g: I\to \mathbb{R}$ are cont. w/ domain interval $I\subseteq \mathbb R$. Then, given $t_0 \in I$ and any $y_0, y_1 \in \mathbb R$ there is a unique func. $y:I\to\mathbb R$ which satisfies:

• $y’’ + py’ + q = g$
• $y(t_0)=y_0 \quad\text{and}\quad y’(t_0)=y_1$

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Wronskian

Sps. $u,v:I\to\mathbb R$ are two diff. func. on interval $I\subseteq \mathbb R$. Then, the Wronskian of the two funcs. is $W:I\to\mathbb R$ s.t.

\(W(t):= \det\begin{bmatrix} u(t) & v(t) \\ u'(t) & v'(t)\end{bmatrix}:=u(t)v'(t)-v(t)u'(t)\)

for all $t\in I$ s.t. if:

• $W(t_0)=0$ then $u,v$ are lin. dep.
• $W(t_0) \not = 0$ then $u,v$ are lin. indep.