Differential Equations

Identify the slope field given specific points and determine which letter corresponds to the correct slope field.

Match the differential equation to its slope field given the options: some equations contain only $x$, some contain only $y$, and some contain both $x$ and $y$.

Solve the higher-order homogeneous linear differential equation using the characteristic equation method. Specifically, consider the equation with constant coefficients and determine the values of R that satisfy the equation such that the solution is linearly independent.

Solve the differential equation $y''' + y' = 0$.

Solve the fourth order constant coefficient differential equation $y^{(4)} - 3y^{(3)} + 3y'' - y' = 0$.

Solve the second order linear homogeneous differential equation: $y'' + 2y' + 8y = 0$ using the quadratic formula to find the characteristic equation roots.

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.

Given the differential equation $y'' + 4y = \, \sin x$, find the general solution.

Find the complementary and particular solutions for the differential equation $y'' + y = \, \cos^2 x$ using the method of undetermined coefficients.

Given a differential equation $\frac{dx}{dt} = F(x)$, determine the stability of its equilibria by graphical analysis.

For a differential equation $\frac{dy}{dt} = f(y)$, determine the stability of the equilibrium solutions using the phase line and the graph of $f(y)$. Define if each equilibrium solution is asymptotically stable, unstable, or semi-stable.

Consider a system where there is no non-conservative work and potential energy is associated with forces in the system. The potential energy is modeled by the function $V(x) = bx^3$. The force is given by $F(x) = -\frac{dV}{dx}$. Calculate the zero points of the force and analyze the stability of these equilibrium points.

Consider the autonomous differential equation $\frac{dx}{dt} = f(x)$, and suppose $x(t) = x^*$ is an equilibrium point; that is, $f(x^*) = 0$. What can we say about the stability of the equilibrium point $x^*$?

For a nonlinear function $f(x)$, explore the stability of equilibria using the derivative of the function at the equilibrium point. Assume $f'(x_e)$ is non-zero.

Solve the homogeneous linear third order differential equation $y''' - 9y'' + 15y' + 25y = 0$.

Solve the third order linear homogeneous differential equation with constant coefficients given by: $r^3 + r^2 - r - 1 = 0$ to find the zeros of the characteristic polynomial.

Solve the homogeneous linear third order differential equation: $y''' - 7y'' - 8y' = 0$.

Given a mass oscillating on a spring with friction, derive the second order constant coefficient differential equation governing its motion.