Internal problem ID [14968]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 3. The method of successive approximation. Exercises page 31
Problem number: 41.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+y^{2}=x^{2}} \] With initial conditions \begin {align*} [y \left (-1\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 55
dsolve([diff(y(x),x)=x^2-y(x)^2,y(-1) = 0],y(x), singsol=all)
\[ y \left (x \right ) = \frac {x \left (\operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \operatorname {BesselK}\left (\frac {3}{4}, \frac {1}{2}\right )-\operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right ) \operatorname {BesselI}\left (-\frac {3}{4}, \frac {1}{2}\right )\right )}{\operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \operatorname {BesselI}\left (-\frac {3}{4}, \frac {1}{2}\right )+\operatorname {BesselK}\left (\frac {3}{4}, \frac {1}{2}\right ) \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )} \]
✓ Solution by Mathematica
Time used: 0.115 (sec). Leaf size: 211
DSolve[{y'[x]==x^2-y[x]^2,{y[-1]==0}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {i \left (x^2 \left (-\operatorname {BesselJ}\left (-\frac {5}{4},\frac {i}{2}\right )+i \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i}{2}\right )+\operatorname {BesselJ}\left (\frac {3}{4},\frac {i}{2}\right )\right ) \operatorname {BesselJ}\left (-\frac {3}{4},\frac {i x^2}{2}\right )+x^2 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {i}{2}\right ) \operatorname {BesselJ}\left (-\frac {5}{4},\frac {i x^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {3}{4},\frac {i}{2}\right ) \left (x^2 \left (-\operatorname {BesselJ}\left (\frac {3}{4},\frac {i x^2}{2}\right )\right )-i \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )\right )\right )}{x \left (2 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {i}{2}\right ) \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )+\left (-\operatorname {BesselJ}\left (-\frac {5}{4},\frac {i}{2}\right )+i \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i}{2}\right )+\operatorname {BesselJ}\left (\frac {3}{4},\frac {i}{2}\right )\right ) \operatorname {BesselJ}\left (\frac {1}{4},\frac {i x^2}{2}\right )\right )} \]