Internal problem ID [13947]
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 (c).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\sin \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-y \,{\mathrm e}^{x}=0} \] With the expansion point for the power series method at \(x = 2\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 509
Order:=6; dsolve(sin(x)*diff(y(x),x$2)+x^2*diff(y(x),x)-exp(x)*y(x)=0,y(x),type='series',x=2);
\[ y \left (x \right ) = \left (1+\frac {\csc \left (2\right ) {\mathrm e}^{2} \left (x -2\right )^{2}}{2}-\frac {{\mathrm e}^{2} \left (4+\cos \left (2\right )-\sin \left (2\right )\right ) \csc \left (2\right )^{2} \left (x -2\right )^{3}}{6}+\frac {\csc \left (2\right )^{3} \left (\left (\frac {3}{2}-\frac {\sin \left (4\right )}{12}-\sin \left (2\right )+\cos \left (2\right )\right ) {\mathrm e}^{2}+\frac {{\mathrm e}^{4} \sin \left (2\right )}{12}\right ) \left (x -2\right )^{4}}{2}+\frac {\left (\left (-210+56 \sin \left (4\right )+\sin \left (6\right )+201 \sin \left (2\right )+\cos \left (6\right )-205 \cos \left (2\right )-6 \cos \left (4\right )\right ) {\mathrm e}^{2}-4 \,{\mathrm e}^{4} \left (-1+\sin \left (4\right )+\cos \left (4\right )+4 \sin \left (2\right )\right )\right ) \csc \left (2\right )^{4} \left (x -2\right )^{5}}{240}\right ) y \left (2\right )+\left (x -2-2 \csc \left (2\right ) \left (x -2\right )^{2}-\frac {\left (-{\mathrm e}^{2} \sin \left (2\right )-4 \cos \left (2\right )+4 \sin \left (2\right )-16\right ) \csc \left (2\right )^{2} \left (x -2\right )^{3}}{6}+\frac {\csc \left (2\right )^{3} \left (\left (-\frac {\cos \left (4\right )}{12}-\frac {2 \sin \left (2\right )}{3}-\frac {\sin \left (4\right )}{12}+\frac {1}{12}\right ) {\mathrm e}^{2}-4 \cos \left (2\right )-\frac {\cos \left (4\right )}{12}+4 \sin \left (2\right )+\frac {\sin \left (4\right )}{3}-\frac {71}{12}\right ) \left (x -2\right )^{4}}{2}+\frac {\left (\left (\left (-12 \cos \left (2\right )-72\right ) \sin \left (2\right )^{2}+108 \sin \left (2\right )+36 \sin \left (4\right )\right ) {\mathrm e}^{2}+2 \sin \left (2\right )^{2} {\mathrm e}^{4}-6 \sin \left (6\right )+817 \cos \left (2\right )+32 \cos \left (4\right )-\cos \left (6\right )-798 \sin \left (2\right )-224 \sin \left (4\right )+832\right ) \csc \left (2\right )^{4} \left (x -2\right )^{5}}{240}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 953
AsymptoticDSolveValue[Sin[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 \csc (2)-13 \cot (2) \csc (2)+12 \cot ^2(2) \csc (2)-12 \cot ^3(2) \csc (2)\right ) (x-2)^5-\frac {1}{20} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)-e^2 \cot ^2(2) \csc (2)\right ) (x-2)^5+\frac {4}{15} \csc (2) \left (3 \csc (2)-4 \cot (2) \csc (2)+4 \cot ^2(2) \csc (2)\right ) (x-2)^5+\frac {1}{6} \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {1}{40} (4 \csc (2)-4 \cot (2) \csc (2)) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5+\frac {2}{5} \csc ^2(2) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5+\frac {1}{40} e^2 \csc (2) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5-\frac {2}{5} \csc ^2(2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^5-\frac {1}{120} e^2 \csc (2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^5+\frac {32}{15} \csc ^4(2) (x-2)^5+\frac {2}{5} e^2 \csc ^3(2) (x-2)^5+\frac {1}{120} e^4 \csc ^2(2) (x-2)^5-\frac {1}{12} \left (3 \csc (2)-4 \cot (2) \csc (2)+4 \cot ^2(2) \csc (2)\right ) (x-2)^4-\frac {1}{12} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^4+\frac {1}{2} \csc (2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^4-\frac {8}{3} \csc ^3(2) (x-2)^4-\frac {1}{3} e^2 \csc ^2(2) (x-2)^4-\frac {1}{6} (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^3+\frac {8}{3} \csc ^2(2) (x-2)^3+\frac {1}{6} e^2 \csc (2) (x-2)^3-2 \csc (2) (x-2)^2+x-2\right )+c_1 \left (-\frac {1}{60} \left (-2 e^2 \csc (2)+4 e^2 \cot (2) \csc (2)-3 e^2 \cot ^2(2) \csc (2)+3 e^2 \cot ^3(2) \csc (2)\right ) (x-2)^5+\frac {1}{15} \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)-e^2 \cot ^2(2) \csc (2)\right ) (x-2)^5-\frac {1}{20} e^2 \csc (2) \left (3 \csc (2)-4 \cot (2) \csc (2)+4 \cot ^2(2) \csc (2)\right ) (x-2)^5-\frac {1}{40} (-4 \csc (2)+4 \cot (2) \csc (2)) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {2}{15} \csc ^2(2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {1}{30} e^2 \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {1}{10} e^2 \csc ^2(2) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5+\frac {1}{15} e^2 \csc ^2(2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^5-\frac {8}{15} e^2 \csc ^4(2) (x-2)^5-\frac {1}{15} e^4 \csc ^3(2) (x-2)^5-\frac {1}{12} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)-e^2 \cot ^2(2) \csc (2)\right ) (x-2)^4+\frac {1}{6} \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^4-\frac {1}{12} e^2 \csc (2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^4+\frac {2}{3} e^2 \csc ^3(2) (x-2)^4+\frac {1}{24} e^4 \csc ^2(2) (x-2)^4-\frac {1}{6} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^3-\frac {2}{3} e^2 \csc ^2(2) (x-2)^3+\frac {1}{2} e^2 \csc (2) (x-2)^2+1\right ) \]