60.4.17 problem 1465

Internal problem ID [11468]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1465
Date solved : Monday, January 27, 2025 at 11:22:27 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 101

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)
 

\[ y = \frac {2 c_3 \left (a^{4}-5 a^{2}+4\right ) {\mathrm e}^{-a x}+2 \left (c_{1} a^{2}+\frac {\sinh \left (3 x \right )}{6}-4 c_{1} -\frac {\cosh \left (3 x \right )}{6}\right ) \left (a +1\right ) \left (a -1\right ) {\mathrm e}^{2 x}+2 c_{2} \left (a^{4}-5 a^{2}+4\right ) {\mathrm e}^{a x}+a^{2} {\mathrm e}^{x}-4 \,{\mathrm e}^{x}+{\mathrm e}^{-x}}{2 a^{4}-10 a^{2}+8} \]

✓ Solution by Mathematica

Time used: 0.064 (sec). Leaf size: 192

DSolve[-Sinh[x] + 2*a^2*y[x] - a^2*D[y[x],x] - 2*D[y[x],{x,2}] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-((a+1) x)} \left (6 \left (a^2-4\right ) e^x \int _1^x\frac {e^{a K[1]} \sinh (K[1])}{2 a^2+4 a}dK[1]+6 \left (a^2-4\right ) e^{2 a x+x} \int _1^x\frac {e^{-a K[2]} \sinh (K[2])}{2 a^2-4 a}dK[2]+6 a^2 c_1 e^x+6 a^2 c_2 e^{(a+3) x}+6 a^2 c_3 e^{2 a x+x}-e^{a x}+3 e^{(a+2) x}-24 c_2 e^{(a+3) x}-24 c_3 e^{2 a x+x}-24 c_1 e^x\right )}{6 \left (a^2-4\right )} \]