Internal problem ID [13628]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\sinh \left (x \right ) y^{\prime \prime }+y^{\prime } x^{2}-{\mathrm e}^{x} y=0} \] With the expansion point for the power series method at \(x = 2\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 278
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);
\[ y \left (x \right ) = \left (1+\frac {{\mathrm e}^{4} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}-\frac {2 \,{\mathrm e}^{2} \left ({\mathrm e}^{2}+4 \,{\mathrm e}^{4}\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {\left ({\mathrm e}^{2}+12 \,{\mathrm e}^{4}+\frac {33 \,{\mathrm e}^{6}}{2}+\frac {{\mathrm e}^{10}}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}-\frac {2 \left ({\mathrm e}^{2}+\frac {53 \,{\mathrm e}^{4}}{2}+98 \,{\mathrm e}^{6}+79 \,{\mathrm e}^{8}+3 \,{\mathrm e}^{10}+\frac {5 \,{\mathrm e}^{12}}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) y \left (2\right )+\left (x -2-\frac {4 \,{\mathrm e}^{2} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}-\frac {2 \,{\mathrm e}^{2} \left (-\frac {31 \,{\mathrm e}^{2}}{2}-\frac {{\mathrm e}^{6}}{2}-4\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {\left (-47 \,{\mathrm e}^{2}-65 \,{\mathrm e}^{4}-{\mathrm e}^{6}-\frac {7 \,{\mathrm e}^{8}}{2}-\frac {7}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}-\frac {2 \left (-\frac {205 \,{\mathrm e}^{2}}{2}-\frac {1537 \,{\mathrm e}^{4}}{4}-\frac {1249 \,{\mathrm e}^{6}}{4}-\frac {85 \,{\mathrm e}^{8}}{4}-17 \,{\mathrm e}^{10}+\frac {{\mathrm e}^{12}}{4}-\frac {{\mathrm e}^{14}}{4}-\frac {11}{4}\right ) {\mathrm e}^{2} \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 931
AsymptoticDSolveValue[Sinh[x]*y''[x]+x^2*y'[x]-Exp[x]*y[x]==0,y[x],{x,2,5}]
\[ 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 ) \]