45.1.2 problem 1. direct method

Internal problem ID [9499]
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 : 1. direct method
Date solved : Tuesday, September 30, 2025 at 06:19:40 PM
CAS classification : [[_Riccati, _special]]

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

With initial conditions

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

✓ Maple. Time used: 0.224 (sec). Leaf size: 89

ode:=diff(y(x),x) = y(x)^2-x; 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {-2 \operatorname {AiryAi}\left (1, x\right ) 3^{{5}/{6}} \pi -3 \operatorname {AiryAi}\left (1, x\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}}-3 \operatorname {AiryBi}\left (1, x\right ) 3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}+2 \operatorname {AiryBi}\left (1, x\right ) \pi 3^{{1}/{3}}}{2 \operatorname {AiryAi}\left (x \right ) 3^{{5}/{6}} \pi +3 \operatorname {AiryAi}\left (x \right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}}+3 \operatorname {AiryBi}\left (x \right ) 3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}-2 \operatorname {AiryBi}\left (x \right ) \pi 3^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.471 (sec). Leaf size: 164

ode=D[y[x],x]==y[x]^2-x; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sqrt [3]{-3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (i x^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} i x^{3/2}\right )-i x^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} i x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )\right )-2 i x^{3/2} \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} i x^{3/2}\right )}{2 x \left (\operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} i x^{3/2}\right )-\sqrt [3]{-3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )\right )} \end{align*}

✗ Sympy

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

TypeError : bad operand type for unary -: list