75.23.2 problem 725

Internal problem ID [17128]
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 : 725
Date solved : Monday, March 31, 2025 at 03:42:40 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime }&=\frac {y-x}{y+x} \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 20

Order:=6; 
ode:=diff(y(x),x) = (y(x)-x)/(x+y(x)); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = 1+x -x^{2}+\frac {4}{3} x^{3}-\frac {5}{2} x^{4}+\frac {16}{3} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 32

ode=D[y[x],x]==(y[x]-x)/(y[x]+x); 
ic={y[0]==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {16 x^5}{3}-\frac {5 x^4}{2}+\frac {4 x^3}{3}-x^2+x+1 \]

✓ Sympy. Time used: 0.854 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = 1 + x - x^{2} + \frac {4 x^{3}}{3} - \frac {5 x^{4}}{2} + \frac {16 x^{5}}{3} + O\left (x^{6}\right ) \]