Question

Use the method of variation of parameters to solve the given initial value problem.\(\mathbf{y}^{\prime}=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right] \mathbf{y}+\left[\begin{array}{l}2 \\\ 1\end{array}\right], \quad \mathbf{y}(0)=\left[\begin{array}{l}0 \\\ 1\end{array}\right]\)

Short answer

Question: Use the method of variation of parameters to solve the following initial value problem: Given the system of linear differential equations: \(\mathbf{y}'=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right] \mathbf{y} + \left[\begin{array}{l}2 \\ 1\end{array}\right]\) with the initial condition: \(\mathbf{y}(0)=\left[\begin{array}{l}0 \\ 1\end{array}\right]\) Find the general solution of this system.

Step-by-step solution

Recall the method of variation of parameters

The method of variation of parameters involves finding a particular solution of the given inhomogeneous system of linear differential equations, then combining it with the complementary solution, which is the solution of the homogeneous system, to find the general solution.

Solve the homogeneous system

First, we need to solve the homogeneous system: \(\mathbf{y}'=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right] \mathbf{y}\) We do this by finding the eigenvalues and eigenvectors of the matrix: \(\det(\mathbf{A} - \lambda \mathbf{I}) = \det\left[\begin{array}{rr}-\lambda & 1 \\ -1 & -\lambda\end{array}\right] = \lambda^2 + 1 = 0\) The eigenvalues are \(\lambda_1 = i\) and \(\lambda_2 = -i\). Next, find the eigenvectors corresponding to each eigenvalue: For \(\lambda_1 = i\): \((\mathbf{A} - i\mathbf{I})\mathbf{v}_1 = \left[\begin{array}{cc}-i & 1 \\ -1 & -i\end{array}\right]\mathbf{v}_1 = \mathbf{0}\) \(\mathbf{v}_1 = \left[\begin{array}{c}1 \\ i\end{array}\right]\) For \(\lambda_2 = -i\): \((\mathbf{A} + i\mathbf{I})\mathbf{v}_2 = \left[\begin{array}{cc}i & 1 \\ -1 & i\end{array}\right]\mathbf{v}_2 = \mathbf{0}\) \(\mathbf{v}_2 = \left[\begin{array}{c}1 \\ -i\end{array}\right]\) The homogeneous solution is then: \(\mathbf{y_h}(t) = \text{Re}\left(c_1 e^{it}\mathbf{v}_1 + c_2 e^{-it}\mathbf{v}_2\right)\)

Find the particular solution using variation of parameters

To find a particular solution to the inhomogeneous system, we write the solution in the following form, replacing the constants \(c_1\) and \(c_2\) with functions \(f(t)\) and \(g(t)\): \(\mathbf{y_p}(t) = \text{Re}\left(\begin{array}{c}f(t)e^{it}+g(t)e^{-it}\\ if(t)e^{it}-ig(t)e^{-it}\end{array}\right)\) To solve for \(f(t)\) and \(g(t)\), we plug this function into the inhomogeneous equation and equate the coefficients of \(e^{it}\) and \(e^{-it}\): \((f'(t)+ig'(t))e^{it} + (g'(t)-if'(t))e^{-it}= \left[\begin{array}{l}2 \\ 1\end{array}\right]\) Differentiating the given function and equating coefficients, we get: \(f'(t)+ig'(t) = 2\) \(g'(t) -if'(t) = 1\) Now, solving for \(f'(t)\) and \(g'(t)\), we get: \(f'(t) = \frac{3}{2}\) \(g'(t) = \frac{1}{2}\) Now, we integrate the above equations to get \(f(t)\) and \(g(t)\): \(f(t) = \frac{3}{2}t + C_1\) \(g(t) = \frac{1}{2}t + C_2\) Using the initial conditions \(\mathbf{y}(0)=\left[\begin{array}{l}0 \\ 1\end{array}\right]\), we can determine the values of \(C_1\) and \(C_2\): \(\left[\begin{array}{l}C_1-C_2\\ iC_1+iC_2\end{array}\right] = \left[\begin{array}{l}0 \\ 1\end{array}\right]\) From this, we get \(C_1 = 0\) and \(C_2 = 0\). So, \(f(t) = \frac{3}{2}t\), \(g(t) = \frac{1}{2}t\)

Combine the complementary and particular solutions

Now, since we have completed finding the homogeneous and particular solutions, we combine them to obtain the general solution: \(\mathbf{y}(t) = \mathbf{y_h}(t) + \mathbf{y_p}(t)\) \(\mathbf{y}(t) = \text{Re}\left(c_1e^{it}\left[\begin{array}{l}1 \\ i\end{array}\right] + c_2e^{-it}\left[\begin{array}{c}1 \\ -i\end{array}\right] + \left[\begin{array}{c}\frac{3}{2}t \\ 1-\frac{1}{2}t\end{array}\right]\right)\) Thus, we have solved the initial value problem using the method of variation of parameters.

Key concepts

Variation of Parameters

The variation of parameters is a powerful technique used to find particular solutions to non-homogeneous differential equations. Unlike the method of undetermined coefficients, it does not assume a specific form for the solution. Instead, it builds on the solutions of the corresponding homogeneous equation.

This approach involves substituting the constants in the homogeneous solution with functions. By doing this, the particular solution is written as a linear combination of the homogeneous solutions, where each constant is replaced with a function. These functions are then determined by plugging back into the original non-homogeneous equation.

In this method, it involves solving a system of equations for these functions, often resulting in integrals that determine the solution. This makes variation of parameters a versatile and widely applicable tool, ideal for equations where the non-homogeneous term is not easily related to the homogeneous solutions. It provides a neat way to deal with complex types of differential equations without the need for guessing.

Initial Value Problem

An initial value problem is a type of differential equation accompanied by specific values of the function(s) at a given point, known as initial conditions. These conditions are crucial, setting the stage to find a unique solution to the differential equation. Initial value problems are common in modeling real-world phenomena like the movement of particles, where the initial state is known.

In the context of the problem presented, the initial value problem is defined by both the differential equation itself and the initial condition \(\mathbf{y}(0) = \begin{bmatrix} 0 \ 1 \end{bmatrix}\). Solving such problems involves finding a function \(\mathbf{y}(t)\) that not only satisfies the differential equation but also fulfills this initial condition at \(t = 0\).

This comprehensive approach ensures that, out of potentially many solutions to the differential equation, the one satisfying the initial condition is accurately captured. Initial value problems form the backbone of many applied mathematics scenarios, providing solutions that align with real-world data.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra, particularly when dealing with linear differential equations. These are used to understand matrices and are pivotal in solving systems of linear differential equations.

Eigenvalues are scalars \(\lambda\) that satisfy the equation \(\mathbf{A} \mathbf{v} = \lambda \mathbf{v}\), where \(\mathbf{v}\) is the eigenvector. This results in a simpler way of handling linear transformations, allowing us to transform complex systems into forms that are much easier to analyze and solve.

In practice, to find eigenvalues, we solve the characteristic equation \(\det(\mathbf{A} - \lambda \mathbf{I}) = 0\). For the given problem, the solution of this equation yielded complex eigenvalues \(i\) and \(-i\), which correspond to a rotation matrix—a common scenario in systems exhibiting oscillatory behavior.

Once eigenvalues are found, eigenvectors, which are the vectors that are transformed only by being scaled along their direction, can be derived. These are crucial for constructing the general solution to the homogeneous differential equation, as they form the basis for representing the system's behavior.