Problem 36 - Advanced Engineering Mathematics

Question

In Problems 35-46, find the general solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=x+y \\ &\frac{d y}{d t}=-2 x-y \end{aligned} $$

Step-by-Step Solution

Answer

The general solution is \( x(t) = c_1 \cos(t) + c_2 \sin(t) \) and \( y(t) = c_1 (\cos(t) + \sin(t)) + c_2 (\sin(t) - \cos(t)) \).

Step 1: Write the System in Matrix Form

The system of differential equations can be represented in matrix form as \( \frac{d}{dt} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 1 & 1 \ -2 & -1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \). Here, the matrix \( A = \begin{bmatrix} 1 & 1 \ -2 & -1 \end{bmatrix} \) represents the coefficients of the system.

Step 2: Find the Eigenvalues of the Coefficient Matrix

To find the eigenvalues of the matrix \( A \), solve the characteristic equation \( \text{det}(A - \lambda I) = 0 \). Computation gives \( \begin{vmatrix} 1 - \lambda & 1 \ -2 & -1 - \lambda \end{vmatrix} = (1 - \lambda)(-1 - \lambda) + 2 = \lambda^2 + 1 \). The eigenvalues are the solutions to \( \lambda^2 + 1 = 0 \), yielding \( \lambda = i \) and \( \lambda = -i \).

Step 3: Find the Eigenvectors for Each Eigenvalue

For \( \lambda = i \), substitute into \( (A - \lambda I)\begin{bmatrix} x_1 \ x_2 \end{bmatrix} = 0 \). This results in solving \( \begin{bmatrix} 1 - i & 1 \ -2 & -1 - i \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \). A valid eigenvector is \( \begin{bmatrix} 1 \ i+1 \end{bmatrix} \).For \( \lambda = -i \), perform the same procedure and obtain an eigenvector \( \begin{bmatrix} 1 \ -i+1 \end{bmatrix} \).

Step 4: Construct the General Solution

The general solution of the system can be found using the real and imaginary parts of the eigenvalues and eigenvectors. It is given by:\[\begin{align*}x(t) &= c_1 e^{0t} \cos(t) + c_2 e^{0t} \sin(t) \y(t) &= c_1 (\cos(t) + \sin(t)) + c_2 (\sin(t) - \cos(t))\end{align*}\]where \( c_1 \) and \( c_2 \) are arbitrary constants determined by initial conditions.

Key Concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts when working with differential equations, especially those involving systems of linear equations. An eigenvalue is a special number, \( \lambda \), which, when multiplied by an identity matrix, results in zero when subtracted from a matrix \( A \) and set into a determinant equation. The corresponding eigenvectors are the non-zero vectors that, when the matrix is applied to them, result in scalar multiples of the original vectors.

In the context of our system, the characteristic equation \( \text{det}(A - \lambda I) = 0 \) helps find the eigenvalues \( \lambda \). Solving this determinant involves calculating the determinant of \( A - \lambda I \). This equation is solved to find the eigenvalues for which there exist non-zero vectors, known as eigenvectors, satisfying \( (A - \lambda I)x = 0 \).

For the given problem, by doing this step-by-step, we found \( \lambda = i \) and \( \lambda = -i \), with corresponding eigenvectors \( \begin{bmatrix} 1 & i+1 \end{bmatrix} \) and \( \begin{bmatrix} 1 & -i+1 \end{bmatrix} \), respectively.

Matrix Form

Transforming a system of differential equations into matrix form simplifies many steps of solving the equations. The coefficients of the variables in the equations are organized into a matrix form, and the variables themselves are represented as vectors.

For our system:

• \( \frac{dx}{dt} = x + y \)
• \( \frac{dy}{dt} = -2x - y \)

This can be rewritten in matrix form as \( \frac{d}{dt} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 1 & 1 \ -2 & -1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \).

Here, the first matrix is the coefficient matrix \( A \), and its elements are derived from the coefficients in the provided differential equations. Once in matrix form, solving such systems generally involves finding eigenvalues and eigenvectors, which guide us to the general solution.

General Solution

The general solution to a system of differential equations is a combination of solutions that fully describe how the variables in the system behave over time. It incorporates arbitrary constants coming from integration, which can be adjusted according to initial or boundary conditions.

With our system solved, identifying the general solution involved using both the eigenvalues and the corresponding eigenvectors. Each solution can be expressed in terms of complex exponentials, which can be transformed into trigonometric functions. This transformation is useful because it turns potentially complex results into more usable real expressions.

For the system we analyzed:

• The expression for \( x(t) \) involves a combination of cosine and sine functions, \( c_1\cos(t) + c_2\sin(t) \).
• The expression for \( y(t) \) includes transformed versions of these solutions, \( c_1(\cos(t) + \sin(t)) + c_2(\sin(t) - \cos(t)) \).

Here, \( c_1 \) and \( c_2 \) are arbitrary constants that are typically determined using initial conditions to satisfy the specific behavior of the solution over time.