Differential Equations

Given a differential equation $\frac{dy}{dx} = \cos(x) - \sin(y)$, if one uses Euler's method starting from $( \pi, \frac{\pi}{4} )$ with a step size of $\frac{\pi}{12}$, what is the approximate value of $y( \frac{5\pi}{6} )$?

Given the differential equation $\frac{dy}{dx} = -xy$, if applying Euler's method starting from $(2,-4)$ with a step size of $.5$, what would be the estimated value for y at x = $2.5$?

What result does applying the Ratio Test to the infinite series $\sum_{n=0}^{\infty} \frac{n!}{(kn)!}$ yield if k is a positive integer greater than one?

Given the differential equation $\frac{dy}{dx} = x^2y$ with initial condition $y(1) = 2$, which step size for Euler's method would most accurately approximate $y(1.2)$?

With the initial condition $(x_0,y_0)=(dollars),(dollars)$ and the differential equation $\frac{dy}{dx}=(dollars)^{(dollars)}$, what will be the new $x$ after two steps using Euler’s method with step size $(dollar)$?

Suppose the series $\sum_{n=2}^{\infty} (-1)^n \cdot \frac{n!}{n^n}$ converges. Which is the best way to determine the nature of its convergence?

If a chemical reaction occurring in a lab is modeled by the differential equation $\frac{dy}{dt} = ky(L-y)$, where $k$ is a constant, $L$ is the limiting amount of product formed, and $y(t)$ is the amount of product at time $t$, how would Euler's method estimate the amount of product formed after 2 hours usin...

For an initial condition given by $(x_0,y_0)=(3,-6)$ and a differential equation $\frac{dy}{dx}=9-x-y$, which pair represents an accurate first approximation using Euler’s method with a step size ($h$) equal to -3?

Given the differential equation $\frac{dy}{dx} = x + y$ with an initial condition of $y(0) = 1$, what is the approximate value of $y(0.2)$ after one iteration of Euler's method with a step size of $0.2$?

Which function do you use in Euler's Method to find the slope at each step?