Internal problem ID [13629]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\sinh \left (x \right ) y^{\prime \prime }+y^{\prime } x^{2}-\sin \left (x \right ) y=0} \] With the expansion point for the power series method at \(x = 2\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 710
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 (x \right ) = \left (1+\frac {\sin \left (2\right ) {\mathrm e}^{2} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}+\frac {\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 ) {\mathrm e}^{2} \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {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 ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}+\frac {2 \left (\left (\left ({\mathrm e}^{2}-2 \,{\mathrm e}^{6}+{\mathrm e}^{10}\right ) \sin \left (2\right )-8 \,{\mathrm e}^{2}-\frac {41 \,{\mathrm e}^{4}}{4}+6 \,{\mathrm e}^{6}+\frac {41 \,{\mathrm e}^{8}}{4}+2 \,{\mathrm e}^{10}+\frac {{\mathrm e}^{12}}{4}-\frac {1}{4}\right ) \cos \left (2\right )+\sin \left (2\right ) \left (\left ({\mathrm e}^{2}+4 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{8}-{\mathrm e}^{10}\right ) \sin \left (2\right )-\frac {33 \,{\mathrm e}^{2}}{2}-\frac {365 \,{\mathrm e}^{4}}{4}-93 \,{\mathrm e}^{6}-\frac {45 \,{\mathrm e}^{8}}{4}+\frac {3 \,{\mathrm e}^{10}}{2}+\frac {{\mathrm e}^{12}}{4}+\frac {1}{4}\right )\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 {\left (\left ({\mathrm e}^{4}-1\right ) \sin \left (2\right )+32 \,{\mathrm e}^{2}+8\right ) {\mathrm e}^{2} \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {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 ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}+\frac {2 \left (\left (-\frac {{\mathrm e}^{10}}{4}+\frac {{\mathrm e}^{6}}{2}-\frac {{\mathrm e}^{2}}{4}\right ) \cos \left (2\right )^{2}+\left (-\frac {3}{4}-5 \,{\mathrm e}^{2}-\frac {3 \,{\mathrm e}^{12}}{4}-5 \,{\mathrm e}^{10}+10 \,{\mathrm e}^{6}+\frac {3 \,{\mathrm e}^{4}}{4}+\frac {3 \,{\mathrm e}^{8}}{4}\right ) \cos \left (2\right )+\left (5 \,{\mathrm e}^{10}-13 \,{\mathrm e}^{2}-27 \,{\mathrm e}^{4}+27 \,{\mathrm e}^{8}+8 \,{\mathrm e}^{6}\right ) \sin \left (2\right )+\frac {11}{4}+\frac {417 \,{\mathrm e}^{2}}{4}+\frac {671 \,{\mathrm e}^{6}}{2}-\frac {31 \,{\mathrm e}^{10}}{4}-\frac {{\mathrm e}^{12}}{4}+\frac {13 \,{\mathrm e}^{8}}{4}+\frac {1609 \,{\mathrm e}^{4}}{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: 934
AsymptoticDSolveValue[Sinh[x]*y''[x]+x^2*y'[x]-Sin[x]*y[x]==0,y[x],{x,2,5}]
\[ 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 ) \]