73.24.15 problem 34.7 (a)

Internal problem ID [15695]
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 : Tuesday, January 28, 2025 at 08:05:28 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*} 1 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 100

Order:=6; 
dsolve(diff(y(x),x$2)-exp(x)*y(x)=0,y(x),type='series',x=1);
 

\[ y = \left (1+\frac {{\mathrm e} \left (x -1\right )^{2}}{2}+\frac {{\mathrm e} \left (x -1\right )^{3}}{6}+\left (\frac {{\mathrm e}^{2}}{24}+\frac {{\mathrm e}}{24}\right ) \left (x -1\right )^{4}+\left (\frac {{\mathrm e}^{2}}{30}+\frac {{\mathrm e}}{120}\right ) \left (x -1\right )^{5}\right ) y \left (1\right )+\left (x -1+\frac {{\mathrm e} \left (x -1\right )^{3}}{6}+\frac {{\mathrm e} \left (x -1\right )^{4}}{12}+\frac {\left (3 \,{\mathrm e}+{\mathrm e}^{2}\right ) \left (x -1\right )^{5}}{120}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

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

AsymptoticDSolveValue[D[y[x],{x,2}]-Exp[x]*y[x]==0,y[x],{x,1,"6"-1}]
 

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