Glossary

When you integrate $\int 3x^2 , dx$, the result is $x^3 + C$, where $C$ is the constant of integration that accounts for any vertical shift of the antiderivative.

The equation $\frac{dy}{dx} = 2x$ is a simple differential equation that describes the rate of change of $y$ with respect to $x$.

The equation $\frac{dy}{dx} = y \sin(x)$ is a first-order differential equation because it only contains the first derivative $\frac{dy}{dx}$.

The general solution to $\frac{dy}{dx} = 2x$ is $y = x^2 + C$, which represents all possible parabolic functions whose derivative is $2x$.

If you are given that $y(0)=4$ for a differential equation, this initial condition allows you to find the specific value of $C$ in the general solution, leading to a unique particular solution.

The equation $\frac{dy}{dx} = \frac{x}{y}$ is a separable differential equation because it can be rearranged to $y , dy = x , dx$.

To solve $\frac{dy}{dx} = x^2y$, you would use separation of variables to rewrite it as $\frac{1}{y} , dy = x^2 , dx$ before integrating.

For the differential equation $\frac{dy}{dx} = 2x$, the function $y = x^2 + 5$ is a solution to a differential equation because its derivative is $2x$.