Internal problem ID [9032]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1453.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _missing_y]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime }-a^{2} y^{\prime }-{\mathrm e}^{2 x a} \left (\sin ^{2}\relax (x )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 230
dsolve(diff(diff(diff(y(x),x),x),x)-a^2*diff(y(x),x)-exp(2*a*x)*sin(x)^2=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {-108 \,{\mathrm e}^{a x} c_{2} a^{8}+108 \,{\mathrm e}^{-a x} c_{1} a^{8}-108 c_{3} a^{9}+9 \,{\mathrm e}^{2 a x} \cos \left (2 x \right ) a^{6}-588 \,{\mathrm e}^{a x} c_{2} a^{6}+588 \,{\mathrm e}^{-a x} c_{1} a^{6}-588 c_{3} a^{7}+33 \,{\mathrm e}^{2 a x} \sin \left (2 x \right ) a^{5}-9 \,{\mathrm e}^{2 a x} a^{6}-36 \,{\mathrm e}^{2 a x} \cos \left (2 x \right ) a^{4}-672 \,{\mathrm e}^{a x} c_{2} a^{4}+672 \,{\mathrm e}^{-a x} c_{1} a^{4}-672 c_{3} a^{5}-12 \,{\mathrm e}^{2 a x} a^{3} \sin \left (2 x \right )-49 \,{\mathrm e}^{2 a x} a^{4}-192 \,{\mathrm e}^{a x} c_{2} a^{2}+192 \,{\mathrm e}^{-a x} c_{1} a^{2}-192 c_{3} a^{3}-56 \,{\mathrm e}^{2 a x} a^{2}-16 \,{\mathrm e}^{2 a x}}{12 \left (9 a^{6}+49 a^{4}+56 a^{2}+16\right ) a^{3}} \]
✓ Solution by Mathematica
Time used: 0.628 (sec). Leaf size: 102
DSolve[-(E^(2*a*x)*Sin[x]^2) - a^2*y'[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{2 a x}}{12 a^3}+\frac {e^{2 a x} \left (\left (4-11 a^2\right ) \sin (2 x)-3 a \left (a^2-4\right ) \cos (2 x)\right )}{4 \left (a^2+1\right ) \left (a^2+4\right ) \left (9 a^2+4\right )}+\frac {c_1 e^{a x}-c_2 e^{-a x}}{a}+c_3 \\ \end{align*}