Differential Equations

Systems of Equations

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Method of Eigenvalues and Eigenvectors

Solved Problems

Example 1.

Find the general solution of the system of differential equations

dxdt=2x+5y,dydt=x+2y.

Solution.

We calculate the eigenvalues λi of the matrix A composed of the coefficients of the given equations:

det(AλI)=|2λ512λ|=0,(2λ)(2λ)5=0,(λ+2)(λ2)5=0,λ29=0,λ1=3,λ2=3.

In this example, the auxiliary equation has two distinct real roots.

Find an eigenvector V1 corresponding to the eigenvalue λ1=3. Substituting λ1=3, we get the vector-matrix equation for V1:

(AλI)V1=0.

Let the eigenvector V1 have components V1=(V11,V21)T (where the superscript T denotes transposition). Then the previous equation can be written as

[235123][V11V21]=0,[5511][V11V21]=0.

After matrix multiplication we get a system of two equations:

{5V11+5V21=0V11V21=0.

Both equations are linearly dependent. From the second equation we find the relation between the coordinates of the eigenvector: V11=V21. Suppose that V21=1. Then V11=1. Thus, the eigenvector V1 has coordinates V1=(1,1)T.

Similarly, we find the second eigenvector V2 corresponding to λ2=3. Let V2= (V21,V22)T. Then

[2+3512+3][V11V21]=0,[1515][V11V21]=0.

We get the system of two identical equations:

{V21+5V22=0V21+5V22=0.

From this we find the coordinates of the eigenvector V2:

V21=5V22,V22=1,V21=5.

Hence, V2=(5,1)T.

Thus, the system has two different eigenvalues and two eigenvectors. The general solution is given by

where are arbitrary numbers.

Example 2.

Find the general solution of the system of differential equations

Solution.

We seek a solution of the form

where is the eigenvalue of the matrix composed of the coefficients of these equations, and is the corresponding eigenvector. Solve the auxiliary equation:

We obtain two eigenvalues as a pair of complex conjugate numbers. Find the eigenvector for the eigenvalue from the following equations:

Both equations are linearly dependent. From the second equation we have:

Thus, the eigenvector is

Consequently, the complex number produces a solution of the form

Transform the exponential function by Euler's formula:

The solution takes the form:

or after multiplication

In the complex solution, the real and imaginary parts are linearly independent. Separating them, we find the general solution:

Thus, the general solution has the form

where are arbitrary numbers.

Example 3.

Find the general solution of the system of differential equations

Solution.

The matrix of the system is diagonal:

Therefore, we can just say that the eigenvectors are

However, we will construct a solution, following the general algorithm. Calculate the eigenvalues of the matrix

The matrix has a single eigenvalue with algebraic multiplicity If we substitute the number into the system of equations for the eigenvector we obtain a singular case:

It is clear that any nonzero vector will be the eigenvector for the given matrix Therefore, it is convenient to take the following two linearly independent vectors as a basis of eigenvectors:

Note that we have the case where the eigenvalue has the same algebraic and geometric multiplicity: This corresponds to the Case in our classification.

The general solution of the system of equations can be written as

Example 4.

Find the general solution of the system of differential equations

Solution.

We calculate the eigenvalues of

Expand the determinant along the first column:

You may notice that one of the roots of the cubic equation is the number Then we get

The quadratic equation, in turn, has roots Therefore, the matrix has three distinct real eigenvalues:

Now we define an eigenvector for each of the eigenvalues.

Determine the vector for the number by solving the vector-matrix equation

Denoting we can write this equation in the form

As a result, we have a system of linear algebraic equations:

In this system, the first and third equations are the same, i.e., the rank of the system is We leave two independent equations and take as a free variable. This yields:

Thus, the eigenvector has coordinates

where for simplicity we set

Similarly, we find the coordinates of the second eigenvector corresponding to the number We put Then we have the following equations:

Let From the third equation we find: Substituting in the first equation, we get:

Hence, the eigenvector is

At one can write:

Now we calculate the coordinates of the third eigenvector corresponding to the number Denoting we obtain the following equations:

We choose as the free variable. From the last equation we express

Substituting in the first equation, we get:

Thus, the eigenvector ({\mathbf{V}_3}) has coordinates

The general solution can be written as

where are arbitrary constants.

Example 5.

Find the general solution of the system

Solution.

Let's start with finding the eigenvalues of the matrix

Factoring the left side, we get

It can be seen that the auxiliary equation has one real and two complex roots (as a pair of complex conjugate numbers):

The eigenvector for the eigenvalue can be found in the same way as in the previous example. The coordinates of are defined by the system of linear equations:

After multiplying we get:

We see that the rank of the system of equations is Therefore, we can choose one independent variable, which we take as Express the other variables in

So the first eigenvector has coordinates

Consider now the pair of complex conjugate roots To find the part of the general solution associated with this pair of roots it is sufficient to take only one number, for example, and find for it the eigenvector which can have complex coordinates. Next, we construct the particular solution of the form

and identify the real and imaginary parts, which will represent two linearly independent solutions. By implementing this plan, we write the matrix-vector equation for the vector

We get the following system of equations:

Transform the first equation into a more convenient form, multiplying it by

Get rid of the complex numbers in the denominators of the coefficients:

Then the first equation becomes:

Let's go back to the system of equations and reduce it to a triangular form to determine its rank:

Transform the second equation:

Here the coefficient of the variable is

Hence, the second equation is

Similarly, we transform the third equation:

Calculate the coefficient of

Then the third equation can be written as

that is, it coincides with the second equation.

So, the rank of the system is and it can be written in the following equivalent form:

We take as an independent variable The other variables and can be successively expressed in terms of

Thus, we determined the eigenvector with complex coordinates:

Now we construct the solution on the basis of eigenvalue and eigenvector and expand it into real and imaginary parts.

Represent by Euler's formula:

Hence,

The calculated real and imaginary parts of the complex vector solution are linearly independent. Taking into account the first component corresponding to the eigenvalue we can write the general real solution of the system as

where are arbitrary numbers.

Example 6.

Find the general solution of the system of differential equations

Solution.

Determine the eigenvalues of the given matrix:

It can be noted that the cubic equation has a root Factoring out the term we obtain:

The roots of the quadratic equation are: Thus, the auxiliary equation is represented as

The initial matrix of the system is symmetric. So it will have three eigenvectors. This means that the algebraic and geometric multiplicity of the root are the same (and equal to ).

Find the eigenvectors corresponding to the number They can be found from the equations

We see that all three equations are identical. Leaving one equation, and choosing as independent variables, we get:

It follows that the coordinates of the first eigenvector (with ) are

Accordingly, the coordinates of the second linearly independent eigenvector (when ) are

Now we define the third eigenvector corresponding to the number

Here we choose as a free variable The other two coordinates are

Hence, the eigenvector has the following coordinates:

The general solution of the system of differential equations is given by

where are arbitrary constants.

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