Solve the initial value problem $\frac{dy}{dt}+3y=6$, $y\left(0\right)=1$.

Find the equilibrium solutions of the equation $z^{\prime }\left(t\right)=z(z-1)(z+6)$ and classify each as stable or unstable.

Solve the initial value problem: $x^{\prime \prime }\left(s\right)=s{e}^s$, $x\left(0\right)=1$, $x^{\prime }\left(0\right)=0$.

For the autonomous differential equation $y^{\prime }\left(t\right)=3y-6$, find the equilibrium solution.

Sketch a direction field for the following differential equation and the solution curve corresponding to the given initial condition.

$y^{\prime }\left(t\right)=5-y$, $y\left(0\right)=7$

Given the differential equation $y^{\prime }=y(5-y)$ and initial value $y\left(0\right)=C$, for what values of $C$ is the solution decreasing for $t\ge 0$?

Solve the initial value problem $y^{\prime }\left(t\right)=\frac{y}{t},y\left(1\right)=2$ for $t>0$.

Express the left-hand side of the equation $x^2{y}^{\prime }\left(x\right)+2xy\left(x\right)=\frac{2x+1}{x}$ as the derivative of a single term, and solve the initial value problem given that $y\left(1\right)=4$.

Find the general solution to the separable equation: $e^{-\frac{w}{4}}{w}^{\prime }\left(s\right)=8s\sin \left(s^2\right)-2s$

Given the differential equation $y^{\prime }\left(t\right)=-2y$, and $y\left(0\right)=4$, with exact solution $y\left(t\right)=4e^{-2t}$, what happens to the global error at $t=0.5$ if the Euler method's time step $\Delta t=0.1$ is halved?

Given $v^{\prime }\left(t\right)=\frac{t}{v}$, and $v\left(0\right)=2$, use Euler's method with $\Delta t=0.4$ to approximate $v\left(1.2\right)$.

Use Euler's method with step size $\Delta x=0.1$ to estimate the value of the solution at $x=0.5$ for the IVP $y^{\prime }=y^2+4$, $y\left(0\right)=0$. Round Euler's approximation to $4$ decimal places. Also, find the exact value $y\left(0.5\right)$.