79.7.1 problem (a)

Internal problem ID [21102]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 2. Theory of First order differential equations. Excercises IV at page 89
Problem number : (a)
Date solved : Thursday, October 02, 2025 at 07:08:21 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&={\mathrm e}^{x}+x \cos \left (y\right ) \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: 0.003 (sec). Leaf size: 18

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

\[ y = x +x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {23}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 29

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

\[ y(x)\to -\frac {23 x^5}{120}-\frac {x^4}{12}+\frac {x^3}{6}+x^2+x \]

✓ Sympy. Time used: 0.399 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = x + x^{2} + \frac {x^{3}}{6} - \frac {x^{4}}{12} - \frac {23 x^{5}}{120} + O\left (x^{6}\right ) \]