73.24.25 problem 34.9 b(i)

Internal problem ID [15705]
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)
Date solved : Tuesday, January 28, 2025 at 08:05:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-{\mathrm e}^{x} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 74

Order:=8; 
dsolve(diff(y(x),x$2)-exp(x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}+\frac {13}{720} x^{6}+\frac {1}{140} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\frac {1}{72} x^{6}+\frac {29}{5040} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 91

AsymptoticDSolveValue[D[y[x],{x,2}]-Exp[x]*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (\frac {29 x^7}{5040}+\frac {x^6}{72}+\frac {x^5}{30}+\frac {x^4}{12}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^7}{140}+\frac {13 x^6}{720}+\frac {x^5}{24}+\frac {x^4}{12}+\frac {x^3}{6}+\frac {x^2}{2}+1\right ) \]