Internal problem ID [13953]
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 (i).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (1-{\mathrm e}^{x}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 49
Order:=6; dsolve(x*diff(y(x),x$2)+(1-exp(x))*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{12} x^{3}+\frac {1}{18} x^{4}+\frac {3}{160} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 63
AsymptoticDSolveValue[x*y''[x]+(1-Exp[x])*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^5}{60}+\frac {x^4}{24}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {3 x^5}{160}+\frac {x^4}{18}+\frac {x^3}{12}+\frac {x^2}{2}+1\right ) \]