Differential Equations

Which method should be used first to solve $\int \sin(y) dy = \int \cos(x) dx$, assuming you have been given an initial condition linking two variables?

If the differential equation $\frac{dy}{dx} = ky(1-\frac{y}{L})$ models a population $P$, where $k$ and $L$ are positive constants, which alteration to the parameters would result in a slower approach to the carrying capacity?

If the differential equation $\frac{dy}{dx} = y^2 \sin(x)$ has a particular solution passing through the point $(\frac{\pi}{2}, k)$, for which value of $k$ does the solution exhibit the steepest tangent line at that point?

A tank contains $Q(t)=100e^{-0.03t}$ grams of a chemical after t hours. If no chemicals are added or removed, which statement best describes how many grams remain after another two hours?

Given the differential equation: $\frac{dy}{dx} = \frac{3}{x}$, find the particular solution for $y(2) = 5$.

How can ignoring domain restrictions when solving differential equations affect the validity of the solutions?

Solve the differential equation: $\frac{dy}{dx} = \frac{1}{x^2}$, given the initial condition $y(1) = 2$.

If the differential equation $\frac{dy}{dx} = y^2 \sin(x)$ is given with the initial condition $y(\pi) = -1$, which of the following functions could be the particular solution?

What role does the constant play in the general solution of a differential equation?

What is the purpose of splitting the domain into intervals or using different methods to obtain solutions for different parts of the domain?