69.23.15 problem 738

Internal problem ID [18499]
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 : 738
Date solved : Thursday, October 02, 2025 at 03:14:23 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&={\mathrm e}^{y}+y x \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 18

Order:=6; 
ode:=diff(y(x),x) = exp(y(x))+x*y(x); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = x +\frac {1}{2} x^{2}+\frac {2}{3} x^{3}+\frac {11}{24} x^{4}+\frac {53}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✗ Mathematica

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

Not solved

✓ Sympy. Time used: 0.221 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) - exp(y(x)) + Derivative(y(x), x),0) 
ics = {y(0): 0} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=6)
 

\[ y{\left (x \right )} = x + \frac {x^{2}}{2} + \frac {2 x^{3}}{3} + \frac {11 x^{4}}{24} + \frac {53 x^{5}}{120} + O\left (x^{6}\right ) \]