73.24.15 problem 34.7 (a)

Internal problem ID [15624]
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)
Date solved : Monday, March 31, 2025 at 01:43:16 PM
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*} 1 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 97

Order:=6; 
ode:=diff(diff(y(x),x),x)-exp(x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = \left (1+\frac {{\mathrm e} \left (x -1\right )^{2}}{2}+\frac {{\mathrm e} \left (x -1\right )^{3}}{6}+\frac {{\mathrm e} \left ({\mathrm e}+1\right ) \left (x -1\right )^{4}}{24}+\frac {{\mathrm e} \left (4 \,{\mathrm e}+1\right ) \left (x -1\right )^{5}}{120}\right ) y \left (1\right )+\left (x -1+\frac {{\mathrm e} \left (x -1\right )^{3}}{6}+\frac {{\mathrm e} \left (x -1\right )^{4}}{12}+\frac {{\mathrm e} \left ({\mathrm e}+3\right ) \left (x -1\right )^{5}}{120}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 121

ode=D[y[x],{x,2}]-Exp[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{30} e^2 (x-1)^5+\frac {1}{120} e (x-1)^5+\frac {1}{24} e^2 (x-1)^4+\frac {1}{24} e (x-1)^4+\frac {1}{6} e (x-1)^3+\frac {1}{2} e (x-1)^2+1\right )+c_2 \left (\frac {1}{120} e^2 (x-1)^5+\frac {1}{40} e (x-1)^5+\frac {1}{12} e (x-1)^4+\frac {1}{6} e (x-1)^3+x-1\right ) \]

✓ Sympy. Time used: 0.757 (sec). Leaf size: 51

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=1,n=6)
 

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