20.24.15 problem Problem 15

Internal problem ID [4000]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.2. page 739
Problem number : Problem 15
Date solved : Monday, January 27, 2025 at 08:05:59 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: 0.003 (sec). Leaf size: 54

Order:=6; 
dsolve(diff(y(x),x$2)-exp(x)*y(x)=0,y(x),type='series',x=0);
 
\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 63

AsymptoticDSolveValue[D[y[x],{x,2}]-Exp[x]*y[x]==0,y[x],{x,0,"6"-1}]
 
\[ y(x)\to c_2 \left (\frac {x^5}{30}+\frac {x^4}{12}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^5}{24}+\frac {x^4}{12}+\frac {x^3}{6}+\frac {x^2}{2}+1\right ) \]