The following example is taken from the book Fundamentals of Differential Equations (by Nagle et al.):
Given the differential equation $$ y''''(x)-k^2y''(x)=q(x),\quad 0<x<1 $$ for the deflection $y(x)$ of an elastic rod under a load $q(x)$ and subject to a constant axial force proportional to $k^2$.
(a) Show that a general solution can be written in the form \begin{align*} y(x)&=C_1+C_2x+C_3e^{kx}+C_4e^{-kx}+\frac{1}{k^2}\int q(x)x\,dx-\frac{x}{k^2}\int q(x)\,dx\\ &\quad +\frac{e^{kx}}{2k^3}\int q(x)e^{-kx}\,dx-\frac{e^{-kx}}{2k^3}\int q(x)e^{kx}\,dx \end{align*} (b) Show that the general solution in (a) can be rewritten in the form \begin{align*} y(x)&=c_1+c_2x+c_3e^{kx}+c_4e^{-kx}\\ &\quad +\frac{1}{k^2}\int_0^xq(s)(s-x)\,ds-\frac{1}{k^3}\int_0^xq(s)\sinh[k(s-x)]\,ds \end{align*} (c) How does the form of the solution in (b) change when the integral is to be taken over the interval $x\leq s\leq 1$ instead?
I know how to answer (a), but (b) and (c) are not clear. Perhaps I am overcomplexifying the problem. Many thanks!
The notation $\int f(x) dx$ denotes a primitive $F(x)$ such that $F'(x)=f(x)$. Similarly, one can write $$F(x)-F(a)=\int_a^x f(s) ds$$ To see this, use the fundamental theorem of calculus, to take the derivative with respect to $x$. $$F'(x)-0=f(x)$$ Now just replace everywhere $\int f(x) dx $ by $\int_0^x f(s) ds$: \begin{align*} y(x)&=C_1+C_2x+C_3e^{kx}+C_4e^{-kx}+\frac{1}{k^2}\int q(x)x\,dx-\frac{x}{k^2}\int q(x)\,dx\\ &\quad +\frac{e^{kx}}{2k^3}\int q(x)e^{-kx}\,dx-\frac{e^{-kx}}{2k^3}\int q(x)e^{kx}\,dx\\&=C_1+C_2x+C_3e^{kx}+C_4e^{-kx}+\frac{1}{k^2}\int_0^x q(s)s\,ds-\frac{x}{k^2}\int_0^x q(s)\,ds\\ &\quad +\frac{e^{kx}}{2k^3}\int_0^x q(s)e^{-ks}\,ds-\frac{e^{-kx}}{2k^3}\int_0^x q(s)e^{ks}\,ds \end{align*} Now grouping the first two integrals and the last two integrals you get \begin{align*} y(x)&=C_1+C_2x+C_3e^{kx}+C_4e^{-kx}+\frac{1}{k^2}\int_0^x q(s)(s-x)\,ds\\ &\quad +\frac{1}{2k^3}\int_0^x q(s)\left(e^{kx}e^{-ks}-e^{-kx}e^{ks}\right)\,ds\end{align*} Then just use the expression for $\sinh$ to get the answer in part b.
For part c, in a similar way, just use https://en.wikipedia.org/wiki/Leibniz_integral_rule#General_form:_Differentiation_under_the_integral_sign