75.1.1 problem 2

Internal problem ID [16664]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 1. Basic concepts and definitions. Exercises page 18
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 09:16:34 AM
CAS classification : [[_Riccati, _special]]

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

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 43

dsolve(diff(y(x),x)=x^2+y(x)^2,y(x), singsol=all)
 

\[ y = -\frac {x \left (\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) c_{1} +\operatorname {BesselY}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right )\right )}{c_{1} \operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )} \]

✓ Solution by Mathematica

Time used: 0.122 (sec). Leaf size: 169

DSolve[D[y[x],x]==x^2+y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {x^2 \left (-2 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {x^2}{2}\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {3}{4},\frac {x^2}{2}\right )-\operatorname {BesselJ}\left (-\frac {5}{4},\frac {x^2}{2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )}{2 x \left (\operatorname {BesselJ}\left (\frac {1}{4},\frac {x^2}{2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )\right )} \\ y(x)\to -\frac {x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {x^2}{2}\right )-x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {x^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )}{2 x \operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )} \\ \end{align*}