Internal problem ID [13639]
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: 34.7 (a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-{\mathrm e}^{x} y=0} \] With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 87
Order:=6; dsolve(diff(y(x),x$2)-exp(x)*y(x)=0,y(x),type='series',x=1);
\[ y \left (x \right ) = \left (1+\frac {{\mathrm e} \left (-1+x \right )^{2}}{2}+\frac {{\mathrm e} \left (-1+x \right )^{3}}{6}+\left (\frac {{\mathrm e}^{2}}{24}+\frac {{\mathrm e}}{24}\right ) \left (-1+x \right )^{4}+\left (\frac {{\mathrm e}^{2}}{30}+\frac {{\mathrm e}}{120}\right ) \left (-1+x \right )^{5}\right ) y \left (1\right )+\left (-1+x +\frac {{\mathrm e} \left (-1+x \right )^{3}}{6}+\frac {{\mathrm e} \left (-1+x \right )^{4}}{12}+\frac {\left ({\mathrm e}^{2}+3 \,{\mathrm e}\right ) \left (-1+x \right )^{5}}{120}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 121
AsymptoticDSolveValue[y''[x]-Exp[x]*y[x]==0,y[x],{x,1,5}]
\[ y(x)\to c_1 \left (\frac {1}{30} e^2 (x-1)^5+\frac {1}{120} e (x-1)^5+\frac {1}{24} e^2 (x-1)^4+\frac {1}{24} e (x-1)^4+\frac {1}{6} e (x-1)^3+\frac {1}{2} e (x-1)^2+1\right )+c_2 \left (\frac {1}{120} e^2 (x-1)^5+\frac {1}{40} e (x-1)^5+\frac {1}{12} e (x-1)^4+\frac {1}{6} e (x-1)^3+x-1\right ) \]