Matrix Theory 9: Matrix Differential Equations

9. Matrix Differential Equations

I. Matrix differentiation and integration

1. Matrix derivative definition: if each element of the matrix is A microfunction of variable t, it is called A (t). Its derivative is defined

As a result, a function can define a higher-order derivative, similarly, a partial derivative.

1. Matrix derivative properties: If A (t) and B (t) are two micro-matrices that can be computed accordingly
   

   (1)

   (2)

   (3)

(4)

(A has nothing to do with t)

This document only certifies

Certificate:

Again

1. Matrix integral definition: if each element of the matrix is A product function on the interval, A (t) is called A product on the interval, and the integral on A (t) is defined
   

   
   

2. Matrix Integral Properties
   

   (1)

   (2)

   (3)

   1. Order Linear homogeneous Constant Coefficient Ordinary Differential Equations
      

   First-order Linear Second Constant Coefficient Ordinary Differential Equations
   

   
   

   In formula, t is the independent variable, and the mona1 function of t is the constant coefficient.
   

   Ling
   

   
   ,

   Then the original equations become the following matrix equations
   

The solution is

Evaluate the solution and verify it.

And t = 0,

It indicates that x (t) is indeed the solution of the equation, and the integral constant is also correct.

 

For example, to solve the differential equations, the initial condition is

Solution :,

Obtain the feature polynomial of ,,

Polynomial defining undetermined coefficients

Solving Equations

1. First-order linear non-homogeneous Constant Coefficient Ordinary Differential Equations
   

Ling

 

 

 

 

 

Convert equations into matrix equations

The constant variation method is used to solve the problem. The solution of homogeneous equations can be set to the solution of non-homogeneous equations,

Substitute the equation:

The credit nature (3) verifies that c (t) is the solution.

With the initial conditions

Note: high-order ordinary differential equations can often be processed as first-order ordinary differential equations,

For example:

, You can get

Generally, Order n ordinary differential equations can be converted into equations composed of n first-order ordinary differential equations.

 

Job: p170-171 5, 9

P177 3, 4