Question

Solve the initial value problem. Eigenpairs of the coefficient matrices were determined in Exercises 1-10.\(\begin{array}{ll}y_{1}^{\prime}=0, & y_{1}(\pi / 2)=-1 \\ y_{2}^{\prime}=3 y_{2}-5 y_{3}, & y_{2}(\pi / 2)=1 \\\ y_{3}^{\prime}=2 y_{2}-3 y_{3}, & y_{3}(\pi / 2)=2\end{array}\)

Short answer

Question: Find the solutions \(y_1(t)\), \(y_2(t)\), and \(y_3(t)\) that satisfy the given system of linear differential equations and the initial conditions: \(\begin{pmatrix}y_1' \\ y_2' \\ y_3' \end{pmatrix} = \begin{pmatrix} 0&0&0 \\ 0&3&-5 \\ 0&2&-3 \end{pmatrix} \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}\) with initial conditions: \( \begin{pmatrix} y_1(\pi/2) \\ y_2(\pi/2) \\ y_3(\pi/2) \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix} \)

Step-by-step solution

Write down the system of linear differential equations

To solve the initial value problem, first we need to express the given equations in matrix form. \(\begin{pmatrix}y_1' \\ y_2' \\ y_3' \end{pmatrix} = \begin{pmatrix} 0&0&0 \\ 0&3&-5 \\ 0&2&-3 \end{pmatrix} \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}\) With the corresponding initial conditions: \( \begin{pmatrix} y_1(\pi/2) \\ y_2(\pi/2) \\ y_3(\pi/2) \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix} \)

Use eigenpairs to find the general solution for the differential equations

Since the eigenpairs of the coefficient matrix is known already, we can directly use these eigenpairs to write the general solution of the system. Let \((\lambda_i, \boldsymbol{v}_i)\) be the eigenpairs of the coefficient matrix, then: \(\begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix} = c_1\boldsymbol{v}_1e^{\lambda_1t} + c_2\boldsymbol{v}_2e^{\lambda_2t} + c_3\boldsymbol{v}_3e^{\lambda_3t}\)

Apply initial conditions to find the constants

To satisfy the given initial conditions, substitute \(t = \pi/2\) into the general solution and set it equal to the initial conditions vector: \(\begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix} = c_1\boldsymbol{v}_1e^{\lambda_1\pi/2} + c_2\boldsymbol{v}_2e^{\lambda_2\pi/2} + c_3\boldsymbol{v}_3e^{\lambda_3\pi/2}\) Now, solve the system of equations for \(c_1\), \(c_2\), and \(c_3\).

Write down the specific solutions

Finally, substitute the constants \(c_1\), \(c_2\), and \(c_3\) found in Step 3 back into the general solution: \(\begin{pmatrix} y_1(t) \\ y_2(t) \\ y_3(t) \end{pmatrix} = c_1\boldsymbol{v}_1e^{\lambda_1t} + c_2\boldsymbol{v}_2e^{\lambda_2t} + c_3\boldsymbol{v}_3e^{\lambda_3t}\) Now we have the solutions \(y_1(t)\), \(y_2(t)\), and \(y_3(t)\) that satisfy both the given differential equations and the initial conditions.

Key concepts

Linear Differential Equations

Linear differential equations are equations that involve an unknown function and its derivatives, which are linear in terms of this function and its derivatives. In the context of our problem, we have a system of linear differential equations. Such a system can be written in the form \

A\mathbf{y'} = B\mathbf{y}\

, where \(A\) is the coefficient matrix, \(\mathbf{y'}\) is the vector of derivatives, and \(\mathbf{y}\) is the vector of unknown functions. The system indicates how each function's rate of change (\