Higher-Order Linear Homogenous

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Table of Contents

• Higher-Order Linear Systems
  • Key Definitions
    • $det(A)=\sum _i^n{a}_{ij}(-1{)}^{i+j}det(cof(A_{ij}))$
  • Steps
  • Solution
    • Auxiliary Functions
      • $x_1(t):=y(t)$
      • $x_1^{\prime }=x_2$
    • General Solution
      • $\overrightarrow{x}(t;C_i)=\sum _i^n{C}_i{e}^{{\lambda }_{i}t}{\overrightarrow{v}}_i$
      • $y(t;C_i)=x_1(t)=\sum _i^n{C}_i{e}^{{\lambda }_{i}t}{\overrightarrow{v}}_{i,1}$
  • General Solution
    • ${\overrightarrow{x}}^{\prime }=A\overrightarrow{x}$
    • $y(t)=\sum _i^n{C}_i{\overrightarrow{y}}_i(t)$

#UCLA #Y1Q3 #Math33B

Higher-Order Linear Systems

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Key Definitions

Limited to homogenous, constant coefficient, linear higher order differentials

Determinant by Laplace (Cofactor) Expansion:

$det(A)=\sum _i^n{a}_{ij}(-1{)}^{i+j}det(cof(A_{ij}))$

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Steps

1. Convert nth order to nxn matrix
2. Solve linear system
3. Convert to linear differential equation

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Solution

Given nth order diff. eq.

Auxiliary Functions

$x_1(t):=y(t)$

S.t.

$x_1^{\prime }=x_2$

and so on.

Then, create a nxn matrix of aux. funcs.:

General Solution

$\overrightarrow{x}(t;C_i)=\sum _i^n{C}_i{e}^{{\lambda }_{i}t}{\overrightarrow{v}}_i$

Such that, we can find the original diff. eq.

$y(t;C_i)=x_1(t)=\sum _i^n{C}_i{e}^{{\lambda }_{i}t}{\overrightarrow{v}}_{i,1}$

E.g.

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General Solution

We can find a solution of form:

${\overrightarrow{x}}^{\prime }=A\overrightarrow{x}$

where A is the companion matrix and if:

So we get the equation in matrix form:

Then, if $y_1,\dots ,y_n$ are solutions to the nth order differential equation, we ca get the vector valued functions: For which, the != 0 Thus the matrix has linearly independent column vectors ${\overrightarrow{y}}_1,\dots ,{\overrightarrow{y}}_n$

Then finally, we get the general solution:

$y(t)=\sum _i^n{C}_i{\overrightarrow{y}}_i(t)$