67.24.25 problem 34.9 b(i)

Internal problem ID [16992]
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 : Thursday, October 02, 2025 at 01:41:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 74

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

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

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

ode=D[y[x],{x,2}]-Exp[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

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

✓ Sympy. Time used: 0.233 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)*exp(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6} e^{3 x}}{720} + \frac {x^{4} e^{2 x}}{24} + \frac {x^{2} e^{x}}{2} + 1\right ) + C_{1} x \left (\frac {x^{4} e^{2 x}}{120} + \frac {x^{2} e^{x}}{6} + 1\right ) + O\left (x^{8}\right ) \]