6.4 Terminology

In Example 6.2 we obtained infinitely many solutions to the differential equation, since we did not specify the constant $c$. We say that $\log|x-2|+c$ is the general solution to the equation and that $\log|x-2|+1$ is the particular solution satisfying $y(1)=1$.

More generally, the methods we will encounter for ‘solving’ differential equations will produce infinitely many possibilities depending on certain constants. If we specify enough initial conditions then these constants are completely determined, and we obtain a unique solution.

Word of caution: in the example in slide 6.3, $y(2)$ is not defined. We say that the solution blows up at $x=2$. Since we have specified the value at $x=1$, this means that our solution only exists in the region $x<2$. In the case of a scientific application, we could interpret this as saying that our model breaks down and is no longer valid once we get close enough to $x=2$.