Internal problem ID [13649]
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: 34.9 b(i).
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 \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 69
Order:=8; dsolve(diff(y(x),x$2)-exp(x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}+\frac {13}{720} x^{6}+\frac {1}{140} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\frac {1}{72} x^{6}+\frac {29}{5040} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 91
AsymptoticDSolveValue[y''[x]-Exp[x]*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {29 x^7}{5040}+\frac {x^6}{72}+\frac {x^5}{30}+\frac {x^4}{12}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^7}{140}+\frac {13 x^6}{720}+\frac {x^5}{24}+\frac {x^4}{12}+\frac {x^3}{6}+\frac {x^2}{2}+1\right ) \]