2nd Order Linear Differentials

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Table of Contents

• 2nd Order Linear Differentials
  • Key Definitions
    • $y_1(t)=\cos \omega t\text{and}{y}_2(t)=\sin \omega t$
    • $y(t)=C_1\cos \omega t+C_2\sin \omega t$
    • $C_1{y}_1+C_2{y}_2:I\to \mathbb{R}$
    • $C_1{y}_1+C_2{y}_2=0$
    • $y(t;C_1,C_2)=C_1{y}_1+C_2{y}_2$
  • Existence and Uniqueness Theorem: 2nd, Linear
  • Wronskian
    • $W(t):=det\left[\begin{array}{cc}u(t)& v(t)\\ u^{\prime }(t)& v^{\prime }(t)\end{array}\right]:=u(t)v^{\prime }(t)-v(t)u^{\prime }(t)$

#UCLA #Y1Q3 #Math33B

2nd Order Linear Differentials

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Key Definitions

Second-Order Linear Differential Equations - diff. eq. of the form: $y^{″}(t)+p(t)y^{\prime }+q(t)y=g(t)$ 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: $y^{″}+{\omega }^{2}y=0$

$y_1(t)=\cos \omega t\text{and}{y}_2(t)=\sin \omega t$

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

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:

• $y^{″}+p{y}^{\prime }+q=g$
• $y(t_0)=y_0\text{and}{y}^{\prime }(t_0)=y_1$

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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.

$W(t):=det\left[\begin{array}{cc}u(t)& v(t)\\ u^{\prime }(t)& v^{\prime }(t)\end{array}\right]:=u(t)v^{\prime }(t)-v(t)u^{\prime }(t)$

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

• $W(t_0)=0$ then $u,v$ are lin. dep.
• $W(t_0)\ne 0$ then $u,v$ are lin. indep.