Answer to Question #178123 in Differential Equations for Rajan Jogani

Question #178123

Find the general solutions of the following system

  1. dx/dt = -4x+2y , dy/dt = 5x/2 + 2y

2. X' = ( -1 3

- 3 5) X

3. X' = ( 4 -5

5 -4)X



1
Expert's answer
2021-04-29T16:43:18-0400

Consider a general system of linear ODE with two variables.

, ,

The solution can be expressed via exponent of the matrix :

, where


2. Let . The characteristic polinomial of A is .

By the Cayley–Hamilton theorem we have . Then

Answer. .


3. Let . The characteristic polinomial of A is

By the Cayley–Hamilton theorem we have , i.e. . Then

Answer.


1. Assume that the matrix A has two different real roots and . The characteristic polinomial of A is . By the Cayley–Hamilton theorem we have . Let and be the eigenvectors of the matrix A, corresponding to the eigenvalues and respectively. They form a basis of the vector space , hence with some unknown coefficients and . Then

. So

Notice that

therefore,

,

and

If we take , then the characteristic polynomial equals to , its roots are and .

implies

Answer.



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