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
      • \(x_1(t):=y(t)\)
      • \(x_1'=x_2\space\)
    • General Solution
      • \(\vec x(t;C_i)=\sum_i^n C_ie^{\lambda_i t}\vec v_i\)
      • \(y(t;C_i)=x_1(t)=\sum_i^n C_ie^{\lambda_i t}\vec v_{i,1}\)
  • General Solution
    • \(\vec x' = A\vec x\)
    • \(y(t)=\sum_i^n C_i\vec y_i(t)\)

#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.

\(x_1'=x_2\space\)

and so on.

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

General Solution

\(\vec x(t;C_i)=\sum_i^n C_ie^{\lambda_i t}\vec v_i\)

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

\(y(t;C_i)=x_1(t)=\sum_i^n C_ie^{\lambda_i t}\vec v_{i,1}\)

E.g.

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General Solution

We can find a solution of form:

\(\vec x' = A\vec x\)

where A is the companion matrix and if:

So we get the equation in matrix form:

Then, if $y_1,…,y_n$ 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 $\vec y_1,…,\vec y_n$

Then finally, we get the general solution:

\(y(t)=\sum_i^n C_i\vec y_i(t)\)