Question

What is the difference between a solution of a differential equation and a solution of an initial-value problem?

Short answer

The differential equation and the initial condition are both satisfied by the only solution, which is $y=x^2$.

Step-by-step solution

Given information

A solution to an initial value problem is the one and only function that solves both the differential equation and the initial value problem, whereas a solution to a differential equation is one of potentially many functions that make the differential equation true.

Calculation

Take the differential equation $\frac{dy}{dx}=2x$, for instance, whose solution is $y=x^2+C$. Therefore, one of the solutions that satisfies the differential equation is $y=x^2+3$. The initial-value problem, however, is$\frac{dy}{dx}=2x;y\left(0\right)=0$. The differential equation and the initial condition are both satisfied by the only solution, which is $y=x^2$.