69.23.14 problem 737

Internal problem ID [18498]
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 : Thursday, October 02, 2025 at 03:14:23 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: 0.007 (sec). Leaf size: 54

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

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

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

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

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

✓ Sympy. Time used: 0.192 (sec). Leaf size: 41

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=6)
 

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