83.14.7 problem 2 (f)

Internal problem ID [22022]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter X. Solution in power series. Ex. XVIII at page 174
Problem number : 2 (f)
Date solved : Thursday, October 02, 2025 at 08:21:45 PM
CAS classification : [[_Riccati, _special]]

\begin{align*} y^{\prime }-y^{2}-x&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 79

Order:=6; 
ode:=diff(y(x),x)-y(x)^2-x = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{5} y \left (0\right )^{6}+x^{4} y \left (0\right )^{5}+x^{3} y \left (0\right )^{4}+\left (x^{2}+\frac {1}{2} x^{5}\right ) y \left (0\right )^{3}+\left (x +\frac {5}{12} x^{4}\right ) y \left (0\right )^{2}+\left (1+\frac {x^{3}}{3}\right ) y \left (0\right )+\frac {x^{2}}{2}+\frac {x^{5}}{20}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 84

ode=D[y[x],x]-y[x]^2-x==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {1}{20} \left (1+20 c_1{}^6+10 c_1{}^3\right ) x^5+\frac {1}{12} \left (12 c_1{}^5+5 c_1{}^2\right ) x^4+\frac {1}{3} \left (3 c_1{}^4+c_1\right ) x^3+\frac {1}{2} \left (1+2 c_1{}^3\right ) x^2+c_1{}^2 x+c_1 \]

✓ Sympy. Time used: 0.280 (sec). Leaf size: 76

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {x^{2} \left (2 C_{1}^{3} + 1\right )}{2} + \frac {x^{5} \left (10 C_{1}^{3} \left (6 C_{1}^{3} + 1\right ) + 20 C_{1}^{3} + 3\right )}{60} + C_{1} + \frac {C_{1} x^{3} \left (3 C_{1}^{3} + 1\right )}{3} + C_{1}^{2} x + \frac {C_{1}^{2} x^{4} \left (12 C_{1}^{3} + 5\right )}{12} + O\left (x^{6}\right ) \]