73.24.4 problem 34.5 (d)

Internal problem ID [15684]
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 (d)
Date solved : Tuesday, January 28, 2025 at 08:04:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 244

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

\[ y = \left (1+\frac {{\mathrm e}^{4} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}-\frac {2 \,{\mathrm e}^{2} \left ({\mathrm e}^{2}+4 \,{\mathrm e}^{4}\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {\left ({\mathrm e}^{2}+12 \,{\mathrm e}^{4}+\frac {33 \,{\mathrm e}^{6}}{2}+\frac {{\mathrm e}^{10}}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}-\frac {2 \,{\mathrm e}^{2} \left ({\mathrm e}^{2}+\frac {53 \,{\mathrm e}^{4}}{2}+98 \,{\mathrm e}^{6}+79 \,{\mathrm e}^{8}+3 \,{\mathrm e}^{10}+\frac {5 \,{\mathrm e}^{12}}{2}\right ) \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) y \left (2\right )+\left (x -2-\frac {4 \,{\mathrm e}^{2} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}-\frac {2 \,{\mathrm e}^{2} \left (-\frac {31 \,{\mathrm e}^{2}}{2}-\frac {{\mathrm e}^{6}}{2}-4\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {\left (-47 \,{\mathrm e}^{2}-65 \,{\mathrm e}^{4}-{\mathrm e}^{6}-\frac {7 \,{\mathrm e}^{8}}{2}-\frac {7}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}-\frac {2 \,{\mathrm e}^{2} \left (-\frac {205 \,{\mathrm e}^{2}}{2}-\frac {1537 \,{\mathrm e}^{4}}{4}-\frac {1249 \,{\mathrm e}^{6}}{4}-\frac {85 \,{\mathrm e}^{8}}{4}-17 \,{\mathrm e}^{10}+\frac {{\mathrm e}^{12}}{4}-\frac {{\mathrm e}^{14}}{4}-\frac {11}{4}\right ) \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 931

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

\[ y(x)\to c_2 \left (-\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}{20} \left (e^2 \coth (2) \text {csch}(2)-e^2 \coth ^2(2) \text {csch}(2)\right ) (x-2)^5+\frac {4}{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}{6} \text {csch}(2) \left (-e^2 \text {csch}(2)+e^2 \coth (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}{5} \text {csch}^2(2) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) (x-2)^5+\frac {1}{40} e^2 \text {csch}(2) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) (x-2)^5-\frac {2}{5} \text {csch}^2(2) (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^5-\frac {1}{120} e^2 \text {csch}(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 {2}{5} e^2 \text {csch}^3(2) (x-2)^5+\frac {1}{120} e^4 \text {csch}^2(2) (x-2)^5-\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}{12} \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^4+\frac {1}{2} \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}{3} e^2 \text {csch}^2(2) (x-2)^4-\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+\frac {1}{6} e^2 \text {csch}(2) (x-2)^3-2 \text {csch}(2) (x-2)^2+x-2\right )+c_1 \left (-\frac {1}{60} \left (e^2 \text {csch}(2)-e^2 \coth (2) \text {csch}(2)-3 e^2 \coth ^2(2) \text {csch}(2)+3 e^2 \coth ^3(2) \text {csch}(2)\right ) (x-2)^5+\frac {1}{15} \text {csch}(2) \left (e^2 \coth (2) \text {csch}(2)-e^2 \coth ^2(2) \text {csch}(2)\right ) (x-2)^5-\frac {1}{20} e^2 \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)) \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^5-\frac {2}{15} \text {csch}^2(2) \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^5-\frac {1}{30} e^2 \text {csch}(2) \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^5-\frac {1}{10} e^2 \text {csch}^2(2) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) (x-2)^5+\frac {1}{15} e^2 \text {csch}^2(2) (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^5-\frac {8}{15} e^2 \text {csch}^4(2) (x-2)^5-\frac {1}{15} e^4 \text {csch}^3(2) (x-2)^5-\frac {1}{12} \left (e^2 \coth (2) \text {csch}(2)-e^2 \coth ^2(2) \text {csch}(2)\right ) (x-2)^4+\frac {1}{6} \text {csch}(2) \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^4-\frac {1}{12} e^2 \text {csch}(2) (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^4+\frac {2}{3} e^2 \text {csch}^3(2) (x-2)^4+\frac {1}{24} e^4 \text {csch}^2(2) (x-2)^4-\frac {1}{6} \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^3-\frac {2}{3} e^2 \text {csch}^2(2) (x-2)^3+\frac {1}{2} e^2 \text {csch}(2) (x-2)^2+1\right ) \]