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ial Equations
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Linear Systems of Differential Equations with Variable Coefficients
A normal linear system of differential equations with variable coefficients can be written as
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erivative
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where xi (t) are unknown functions, which are continuous and differentiable on an interval [a, b]. The coeffi and the free terms fi (t) are continuous functions on the interval [a, b]. Using vector-matrix notation, this system of equations can be written as
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where
In the general case, the matrix A(t) and the vector functions X(t), f(t) can take both real and complex values. The corresponding homogeneous system with variable coefficients in vector form is given by
Fundamental System of Solutions and Fundamental Matrix
The vector functions x1(t), x2(t), ..., xn(t) are linearly dependent on the interval [a, b], if there are numbers c1 not all zero, such that the following identity holds:
If this identity is satisfied only if
the vector functions xi (t) are called linearly independent on the given interval.
Any system of n linearly independent solutions x1(t), x2(t), ..., xn(t) is called a fundamental system of solutio
A square matrix Φ(t) whose columns are formed by linearly independent solutions x1(t), x2(t), ..., xn(t) is cal fundamental matrix of the system of equations. It has the following form:
where xij (t) are the coordinates of the linearly independent vector solutions x1(t), x2(t), ..., xn(t).
Note that the fundamental matrix Φ(t) is nonsingular, i.e. there always exists the inverse matrix Φ −1(t). Sinc fundamental matrix has n linearly independent solutions, after its substitution into the homogeneous system the identity
We multiply this equation on the right by the inverse function Φ −1(t):
The resulting relation uniquely defines a homogeneous system of equations, given the fundamental matrix.
The general solution of the homogeneous system is expressed in terms of the fundamental matrix in the form
where C is an n-dimensional vector consisting of arbitrary numbers.
Let us mention an interesting special case of homogeneous systems. It turns out that if the product of the ma and the integral of this matrix is commutative, i.e.
the fundamental matrix Φ(t) for such a system of equations is given by
Such property is satisfied in the case of symmetric matrices and, in particular, in the case of diagonal matric
Wronskian and Liouville's Formula
The determinant of the fundamental matrix Φ(t) is called the Wronskian of the system of solutions x1(t), x2(t
The Wronskian is useful to check the linear independence of solutions. The following rules apply:
The solutions x1(t), x2(t), ..., xn(t) of the homogeneous system form a fundamental system if and only corresponding Wronskian is not zero at any point t of the interval [a, b].
The solutions x1(t), x2(t), ..., xn(t) are linearly dependent on the interval [a, b] if and only if the Wron identically zero on this interval.
The Wronskian of the solutions x1(t), x2(t), ..., xn(t) is given by Liouville's formula:
where tr (A(τ)) is the trace of the matrix A(τ), i.e. the sum of all diagonal elements:
Liouville's formula can be used to construct the general solution of the homogeneous system if a particular s known.
Method of Variation of Constants (Lagrange Method)
Now we consider the nonhomogeneous systems that can be written in the vector-matrix form as
The general solution of such a system is the sum of the general solution X0(t) of the corresponding homogen system and a particular solution X1(t) of the nonhomogeneous system, i.e.
where Φ(t) is a fundamental matrix, C is an arbitrary vector.
The most common method for solving the nonhomogeneous systems is the method of variation of constants method). With this method, instead of the constant vector C, we consider the vector C(t) whose components continuously differentiable functions of the independent variable t, i.e. we assume
Substituting this into the nonhomogeneous system, we find the unknown vector C(t):
Given that the matrix Φ(t) is nonsingular, we multiply this equation on the left by Φ−1(t):
After integration we obtain the vector C(t). Example 1
Write the linear system of equations with the following solutions:
Solution.
In the problem the fundamental matrix of the system is known:
We compute the inverse matrix Φ−1(t):
Here Cij denote the cofactors of the corresponding elements of the fundamental matrix Φ(t). The coefficient matrix of the system of equations is given by
The derivative of the fundamental matrix (it is calculated element by element) is equal to
Hence, we obtain:
Thus, the system of equations whose solutions are x1(t), x2(t), can be written as
Example 2
Find a fundamental matrix of the system of differential equations
making sure that the coefficient matrix A(t) commutes with its integral. Solution.
We first show that the multiplication of the matrix A(t) by its integral is commutative. The original matrix is
The integral of the matrix A(t) is found by elementwise integration. For simplicity, we take the lower bound integration to be zero. Then
As a result, we have
So, the commutative property of the matrix product is true. Therefore, the fundamental matrix is given by
We compute the matrix exponential by converting the matrix to diagonal form. In this case, the eigenvalues d the variable t and can be expressed as follows:
For each eigenvalue, we find the corresponding eigenvector. For λ1 we obtain:
Similarly, we find the eigenvector V2 = (V12, V22) T for the eigenvalue λ2:
Then the matrix of reduction to diagonal form (more precisely to Jordan form) is given by
We compute the inverse matrix H −1:
Hence, the Jordan form J is as follows:
The exponential of the matrix J is given by
We can now calculate the fundamental matrix Φ(t):
Example 3
Find the general solution of the system
if one solution is known:
Solution.
Let the second linearly independent solution be expressed by the vector function
with the initial condition
.
We then use Liouville's formula, which is written as
Hence we obtain the relation between the unknown functions u and v:
Consider the second equation of the original system. Substituting the solution X2(t), we write it in the form o
From the previous equation, we can express the term tv:
Substitute it into the differential equation for the function v(t):
Given that tu = v − 1, we obtain a first order linear equation for the function v(t):
We first find the solution of the corresponding homogeneous equation.
where C is an arbitrary number.
Now we determine the solution of the nonhomogeneous equation, using the method of variation of paramet
After substitution we get the expression for the derivative dC/dt:
Integrating, we find the function C(t):
Then the function v(t) will be expressed by the formula
Further, it is easy to find the function u(t):
So the second solution of the system is
The general solution is written as
where C1, C2 are arbitrary constants. Example 4
Find the general solution of the nonhomogeneous system of equations:
Solution.
First we construct the general solution of the homogeneous system
Note that the matrix A(t) of the system is symmetric. Check the product of the matrix A(t) and its integral (th integration is performed element by element) for commutativity.
As can be seen, the multiplication of the matrices is commutative. Therefore, the fundamental matrix of the given by
Now we perform the necessary transformations with the matrix exponential to write the general solution of homogeneous system.
Find the eigenvalues:
For each eigenvalue λ1, λ2, we determine the corresponding eigenvectors. For the value λ1 we obtain:
Similarly we find the eigenvector V2 = (V12, V22) T for the number λ2:
Hence, the transition matrix from the original matrix A(t) to the Jordan form J is given by
Compute the inverse matrix H −1:
Let us see that the Jordan form J of the matrix A(t) is a diagonal matrix with the eigenvalues λ1, λ2 on the dia
The matrix exponential of the matrix J is equal to
Then the fundamental matrix Φ(t) takes the following form:
Thus, the general solution of the homogeneous system is given by
Now we find a particular solution X1(t) of the nonhomogeneous system. In accordance with the method of v parameters, we replace the constant vector C with the vector function C(t). The derivative of this function i the relation
Compute the inverse matrix Φ−1(t) in this formula.
As a result, we obtain the following expression for the derivative C'(t):
Integrating this expression gives:
The last integral is calculated by parts:
Hence,
where A1 is an arbitrary constant. Similarly we find the function C2(t):
Thus, the general solution of the original nonhomogeneous system is written as
Here the component X1(t) corresponding to the inhomogeneous term of the system allows for a simpler repr
Thus, the final answer is
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