51.1.5 problem 3. series method

Internal problem ID [8216]
Book : A course in Ordinary Differential Equations. by Stephen A. Wirkus, Randall J. Swift. CRC Press NY. 2015. 2nd Edition
Section : Chapter 8. Series Methods. section 8.2. The Power Series Method. Problems Page 603
Problem number : 3. series method
Date solved : Sunday, March 30, 2025 at 12:48:49 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=y+x \,{\mathrm e}^{y} \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: 0.006 (sec). Leaf size: 20

Order:=8; 
ode:=diff(y(x),x) = y(x)+x*exp(y(x)); 
ic:=y(0) = 0; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = \frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}+\frac {43}{720} x^{6}+\frac {151}{5040} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 46

ode=D[y[x],x]==y[x]+x*Exp[y[x]]; 
ic={y[0]==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {151 x^7}{5040}+\frac {43 x^6}{720}+\frac {x^5}{15}+\frac {x^4}{6}+\frac {x^3}{6}+\frac {x^2}{2} \]

✓ Sympy. Time used: 1.000 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*exp(y(x)) - 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=8)
 

\[ y{\left (x \right )} = \frac {x^{2}}{2} + \frac {x^{3}}{6} + \frac {x^{4}}{6} + \frac {x^{5}}{15} + \frac {43 x^{6}}{720} + \frac {151 x^{7}}{5040} + O\left (x^{8}\right ) \]