75.23.14 problem 737

Internal problem ID [17211]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.1 Integration of differential equation in series. Power series. Exercises page 171
Problem number : 737
Date solved : Tuesday, January 28, 2025 at 09:57:42 AM
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*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 54

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

\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 63

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

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