Internal problem ID [9042]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1465.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }-a^{2} y^{\prime }+2 a^{2} y-\sinh \left (x \right )=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 114
dsolve(diff(diff(diff(y(x),x),x),x)-2*diff(diff(y(x),x),x)-a^2*diff(y(x),x)+2*a^2*y(x)-sinh(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {6 a^{3} {\mathrm e}^{x}+2 \,{\mathrm e}^{2 x} \sinh \left (3 x \right ) a^{3}-2 \,{\mathrm e}^{2 x} \cosh \left (3 x \right ) a^{3}-24 \,{\mathrm e}^{x} a -2 \,{\mathrm e}^{2 x} \sinh \left (3 x \right ) a +2 \,{\mathrm e}^{2 x} \cosh \left (3 x \right ) a +6 a \,{\mathrm e}^{-x}}{12 a \left (a^{2}-4\right ) \left (a -1\right ) \left (a +1\right )}+c_{1} {\mathrm e}^{2 x}+c_{2} {\mathrm e}^{a x}+c_{3} {\mathrm e}^{-a x} \]
✓ Solution by Mathematica
Time used: 0.076 (sec). Leaf size: 48
DSolve[-Sinh[x] + 2*a^2*y[x] - a^2*y'[x] - 2*y''[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \sinh (x)+\cosh (x)}{3 a^2-3}+c_1 e^{-a x}+c_3 e^{a x}+c_2 e^{2 x} \\ \end{align*}