Worked exercise — y' = −2xy (the Gaussian)

Problem

Solve $y'=-2xy$ (separable: the solution is a Gaussian!).

Solution

We separate the variables: $\dfrac{dy}{y} = -2x\,dx$. Integrating: $\ln|y| = -x^2 + C$, from which: $\boxed{y(x) = A\,e^{-x^2}}$ This is exactly the shape of the Gaussian bell curve.

Links

Topics: Differential equations
Concepts: Separable variables
Functions: Exponential function
Skills: Integrating · Solving equations
Exercise type: Equation solving