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 : Monday, January 27, 2025 at 03:46:11 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*}

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{D[y[x],x]==y[x]+x*Exp[y[x]],{y[0]==0}},y[x],{x,0,"8"-1}]
 

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