60.4.5 problem 1453

Internal problem ID [11456]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1453
Date solved : Monday, January 27, 2025 at 11:22:21 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }-a^{2} y^{\prime }-{\mathrm e}^{2 a x} \sin \left (x \right )^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 124

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)
 

\[ y = \frac {\left (\left (-9 a^{6}+36 a^{4}\right ) \cos \left (2 x \right )+\left (-33 a^{5}+12 a^{3}\right ) \sin \left (2 x \right )+9 a^{6}+49 a^{4}+56 a^{2}+16\right ) {\mathrm e}^{2 a x}+108 \left (a^{2}+4\right ) \left (a c_3 +c_{1} {\mathrm e}^{a x}-c_{2} {\mathrm e}^{-a x}\right ) a^{2} \left (a^{2}+1\right ) \left (a^{2}+\frac {4}{9}\right )}{108 \left (a^{2}+4\right ) a^{3} \left (a^{2}+1\right ) \left (a^{2}+\frac {4}{9}\right )} \]

✓ Solution by Mathematica

Time used: 22.071 (sec). Leaf size: 97

DSolve[-(E^(2*a*x)*Sin[x]^2) - a^2*D[y[x],x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^xe^{-a K[3]} \left (e^{2 a K[3]} c_1+c_2+e^{2 a K[3]} \int _1^{K[3]}\frac {e^{a K[1]} \sin ^2(K[1])}{2 a}dK[1]+\int _1^{K[3]}-\frac {e^{3 a K[2]} \sin ^2(K[2])}{2 a}dK[2]\right )dK[3]+c_3 \]