Differential Equations: Systems of Linear Differential Equations

Solve a system of linear first-order differential equations using matrix methods.

Use the Matrix method to solve this linear system of differential equations: rac{dX}{dt} = 6x + 5y and rac{dY}{dt} = x + 2y.

Rewrite the second order differential equation as a system of first-order linear differential equations.

Rewrite the fourth order differential equation as a system of first-order linear differential equations.

Solve the system by transforming it into a single differential equation: $X' = -2Y, Y' = X$.

Solve the system by transforming it into a single differential equation: $X' = Y, Y' = 6X - Y$.

Find the fundamental matrix for the system $X' = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} X$.

Find a fundamental matrix for a given constant coefficient ODE system, which you have solved before in the last couple sections.

Consider the system $X' = AX$, where $A = \begin{pmatrix} 5 & -1 \\ -2 & 1 \end{pmatrix}$. Find a fundamental matrix $\Phi(t)$ such that $\Phi(0)$ is the identity matrix and use it to solve the initial value problem with $X(0) = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$.

Convert the nth order linear differential equation to a system of linear equations in normal form.

Given a third order differential equation, reduce it to a system of three first order differential equations using variables $X_1$, $X_2$, and $X_3$ where $X_1 = x$, $X_2 = x'$, and $X_3 = x''$.

Solve the given system of differential equations using the eigenvalue method: \begin{align*} 1 &= 2x + 3y \\ 7 &= 4x + y \end{align*}

Given a system of linear first-order differential equations with complex conjugate eigenvalues in matrix form, find the general solution using the eigenvalue method and the principle of superposition.

Given a system of two first-order, linear, homogeneous differential equations with distinct real eigenvalues, determine the eigenvalues and eigenvectors, and write down the general solution.