73.24.5 problem 34.5 (e)

Internal problem ID [15685]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.5 (e)
Date solved : Tuesday, January 28, 2025 at 08:04:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 458

Order:=6; 
dsolve(sinh(x)*diff(y(x),x$2)+x^2*diff(y(x),x)-sin(x)*y(x)=0,y(x),type='series',x=2);
 

\[ y = \left (1+\frac {\sin \left (2\right ) {\mathrm e}^{2} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}+\frac {{\mathrm e}^{2} \left (\left (-8 \,{\mathrm e}^{2}-{\mathrm e}^{4}-1\right ) \sin \left (2\right )+\cos \left (2\right ) \left ({\mathrm e}^{4}-1\right )\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {2 \,{\mathrm e}^{2} \left (\frac {\left (-{\mathrm e}^{2}+{\mathrm e}^{6}\right ) \sin \left (2\right )^{2}}{4}+\left (5 \,{\mathrm e}^{2}+9 \,{\mathrm e}^{4}+{\mathrm e}^{6}\right ) \sin \left (2\right )+\cos \left (2\right ) \left ({\mathrm e}^{2}-{\mathrm e}^{6}-\frac {{\mathrm e}^{8}}{4}+\frac {1}{4}\right )\right ) \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}+\frac {4 \,{\mathrm e}^{2} \left (\left (-\frac {1}{8}+3 \,{\mathrm e}^{6}+\frac {41 \,{\mathrm e}^{8}}{8}-4 \,{\mathrm e}^{2}+{\mathrm e}^{10}+\frac {{\mathrm e}^{12}}{8}-\frac {41 \,{\mathrm e}^{4}}{8}\right ) \cos \left (2\right )+\frac {\left (\frac {1}{4}-93 \,{\mathrm e}^{6}-\frac {33 \,{\mathrm e}^{2}}{2}-\frac {45 \,{\mathrm e}^{8}}{4}+\frac {3 \,{\mathrm e}^{10}}{2}+\frac {{\mathrm e}^{12}}{4}-\frac {365 \,{\mathrm e}^{4}}{4}\right ) \sin \left (2\right )}{2}+\frac {\left (-\cos \left (4\right )+\sin \left (4\right )+1\right ) {\mathrm e}^{2}}{4}+\frac {\left (\cos \left (4\right )+\sin \left (4\right )-1\right ) {\mathrm e}^{10}}{4}+\left (1-\cos \left (4\right )\right ) {\mathrm e}^{4}-\frac {{\mathrm e}^{6} \sin \left (4\right )}{2}+{\mathrm e}^{8} \left (-1+\cos \left (4\right )\right )\right ) \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) y \left (2\right )+\left (x -2-\frac {4 \,{\mathrm e}^{2} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}+\frac {{\mathrm e}^{2} \left (\left ({\mathrm e}^{4}-1\right ) \sin \left (2\right )+32 \,{\mathrm e}^{2}+8\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {2 \,{\mathrm e}^{2} \left (-\frac {7}{4}+\left (2 \,{\mathrm e}^{2}-2 \,{\mathrm e}^{6}-\frac {{\mathrm e}^{8}}{4}+\frac {1}{4}\right ) \sin \left (2\right )+\frac {\left (\frac {1}{2}-{\mathrm e}^{4}+\frac {{\mathrm e}^{8}}{2}\right ) \cos \left (2\right )}{2}-24 \,{\mathrm e}^{2}-\frac {69 \,{\mathrm e}^{4}}{2}+\frac {{\mathrm e}^{8}}{4}\right ) \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}-\frac {{\mathrm e}^{2} \left (-11+\left ({\mathrm e}^{10}+{\mathrm e}^{2}\right ) \cos \left (2\right )^{2}+\left (20 \,{\mathrm e}^{2}-3 \,{\mathrm e}^{4}-40 \,{\mathrm e}^{6}-3 \,{\mathrm e}^{8}+20 \,{\mathrm e}^{10}+3 \,{\mathrm e}^{12}+3\right ) \cos \left (2\right )+4 \left (13 \,{\mathrm e}^{2}+27 \,{\mathrm e}^{4}-8 \,{\mathrm e}^{6}-27 \,{\mathrm e}^{8}-5 \,{\mathrm e}^{10}\right ) \sin \left (2\right )+31 \,{\mathrm e}^{10}-417 \,{\mathrm e}^{2}-1609 \,{\mathrm e}^{4}+\left (-\cos \left (4\right )-1343\right ) {\mathrm e}^{6}+{\mathrm e}^{12}-13 \,{\mathrm e}^{8}\right ) \left (x -2\right )^{5}}{30 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 934

AsymptoticDSolveValue[Sinh[x]*D[y[x],{x,2}]+x^2*D[y[x],x]-Sin[x]*y[x]==0,y[x],{x,2,"6"-1}]
 

\[ y(x)\to c_2 \left (-\frac {1}{20} \left (\cos (2) \coth (2) \text {csch}(2)+\text {csch}(2) \sin (2)-\coth ^2(2) \text {csch}(2) \sin (2)\right ) (x-2)^5+\frac {1}{6} \text {csch}(2) (-\cos (2) \text {csch}(2)+\coth (2) \text {csch}(2) \sin (2)) (x-2)^5+\frac {1}{120} \text {csch}^2(2) \sin ^2(2) (x-2)^5+\frac {1}{40} \text {csch}(2) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) \sin (2) (x-2)^5-\frac {1}{120} \text {csch}(2) (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) \sin (2) (x-2)^5+\frac {2}{5} \text {csch}^3(2) \sin (2) (x-2)^5-\frac {1}{60} \left (-6 \text {csch}(2)+7 \coth (2) \text {csch}(2)+12 \coth ^2(2) \text {csch}(2)-12 \coth ^3(2) \text {csch}(2)\right ) (x-2)^5+\frac {1}{15} \text {csch}(2) \left (-\text {csch}(2)-4 \coth (2) \text {csch}(2)+4 \coth ^2(2) \text {csch}(2)\right ) (x-2)^5-\frac {1}{5} \text {csch}(2) \left (\text {csch}(2)+4 \coth (2) \text {csch}(2)-4 \coth ^2(2) \text {csch}(2)\right ) (x-2)^5-\frac {1}{40} (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) (x-2)^5+\frac {2}{3} \text {csch}^2(2) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) (x-2)^5-\frac {2}{15} \text {csch}^2(2) (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^5+\frac {32}{15} \text {csch}^4(2) (x-2)^5-\frac {1}{12} (-\cos (2) \text {csch}(2)+\coth (2) \text {csch}(2) \sin (2)) (x-2)^4-\frac {1}{3} \text {csch}^2(2) \sin (2) (x-2)^4-\frac {1}{12} \left (-\text {csch}(2)-4 \coth (2) \text {csch}(2)+4 \coth ^2(2) \text {csch}(2)\right ) (x-2)^4-\frac {1}{3} \text {csch}(2) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) (x-2)^4+\frac {1}{6} \text {csch}(2) (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^4-\frac {8}{3} \text {csch}^3(2) (x-2)^4+\frac {1}{6} \text {csch}(2) \sin (2) (x-2)^3-\frac {1}{6} (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^3+\frac {8}{3} \text {csch}^2(2) (x-2)^3-2 \text {csch}(2) (x-2)^2+x-2\right )+c_1 \left (-\frac {1}{60} \left (2 \cos (2) \text {csch}(2)-3 \cos (2) \coth ^2(2) \text {csch}(2)-4 \coth (2) \text {csch}(2) \sin (2)+3 \coth ^3(2) \text {csch}(2) \sin (2)\right ) (x-2)^5+\frac {1}{15} \text {csch}(2) \left (\cos (2) \coth (2) \text {csch}(2)+\text {csch}(2) \sin (2)-\coth ^2(2) \text {csch}(2) \sin (2)\right ) (x-2)^5-\frac {1}{30} \text {csch}(2) \sin (2) (-\cos (2) \text {csch}(2)+\coth (2) \text {csch}(2) \sin (2)) (x-2)^5-\frac {1}{40} (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) (-\cos (2) \text {csch}(2)+\coth (2) \text {csch}(2) \sin (2)) (x-2)^5-\frac {2}{15} \text {csch}^2(2) (-\cos (2) \text {csch}(2)+\coth (2) \text {csch}(2) \sin (2)) (x-2)^5-\frac {1}{15} \text {csch}^3(2) \sin ^2(2) (x-2)^5+\frac {1}{20} \text {csch}(2) \left (\text {csch}(2)+4 \coth (2) \text {csch}(2)-4 \coth ^2(2) \text {csch}(2)\right ) \sin (2) (x-2)^5-\frac {1}{6} \text {csch}^2(2) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) \sin (2) (x-2)^5-\frac {8}{15} \text {csch}^4(2) \sin (2) (x-2)^5-\frac {1}{12} \left (\cos (2) \coth (2) \text {csch}(2)+\text {csch}(2) \sin (2)-\coth ^2(2) \text {csch}(2) \sin (2)\right ) (x-2)^4+\frac {1}{6} \text {csch}(2) (-\cos (2) \text {csch}(2)+\coth (2) \text {csch}(2) \sin (2)) (x-2)^4+\frac {1}{24} \text {csch}^2(2) \sin ^2(2) (x-2)^4+\frac {1}{12} \text {csch}(2) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) \sin (2) (x-2)^4+\frac {2}{3} \text {csch}^3(2) \sin (2) (x-2)^4-\frac {1}{6} (-\cos (2) \text {csch}(2)+\coth (2) \text {csch}(2) \sin (2)) (x-2)^3-\frac {2}{3} \text {csch}^2(2) \sin (2) (x-2)^3+\frac {1}{2} \text {csch}(2) \sin (2) (x-2)^2+1\right ) \]