Q:

In a general solution to the differential equation, are each of the individual terms continuous?

Given some complete general solution to an Nth order differential equation:
$$y(x) = c_1f_1(x) + c_2f_2(x) + c_3f_3(x) + c_4f_4(x) + ... c_n f_n(x)$$
Must each of the terms $f_i(x)$ be continuous? If so, could you please explain why? I wish to use certain methods of solving differential equations (by finding a piece wise constant $c_n$ that makes y continuous after a faulty integral), which rely upon the fact that all of the (n-1)TH derivatives of y are continuous (because then I have n linear equations), but if each of the terms are all that need to be continuous then it is much easier to solve.

A:

We say a linear $n$th order ODE is in normal form if it is written:
$$\sum_{i=0}^{n-1} a_i(t) y^{(i)}(t) + y^{(n)}(t) = f(t)$$
i.e. the leading coefficient is $1$. When all the $a_i$ are continuous and $f$ is continuous, a strong solution exists and in particular is $C^n$. If the $a_i$ are not continuous but are locally bounded and have only jump discontinuities, then a weak solution exists and is $C^{n-1}$. More serious problems can arise when the coefficients are not locally bounded. (I'm not sure what happens if the $a_i$ have essential discontinuities.)
These problems are most easily exemplified by the equation
$$y'+\frac{1}{x}y=0$$
whose nontrivial solutions all blow up at $x=0$. This equation is a good example of why we have to consider the normal form, because one can write this as $xy'+y=0$, in which form the coefficients are continuous.

