75.23.15 problem 738

Internal problem ID [17212]
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 : Tuesday, January 28, 2025 at 09:57:43 AM
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*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 18

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

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

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 33

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

\[ y(x)\to \frac {53 x^5}{120}+\frac {11 x^4}{24}+\frac {2 x^3}{3}+\frac {x^2}{2}+x \]