Internal problem ID [12232]
Book: DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR
PUBLISHERS, Moscow 1969.
Section: Chapter 8. Differential equations. Exercises page 595
Problem number: 194.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Riccati, _special]]
\[ \boxed {y^{\prime }-y^{2}=x} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 97
dsolve([diff(y(x),x)=y(x)^2+x,y(0) = 1],y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-2 \pi 3^{\frac {5}{6}}+3 \,3^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )^{2}\right ) \operatorname {AiryAi}\left (1, -x \right )+\operatorname {AiryBi}\left (1, -x \right ) \left (3 \,3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}+2 \pi 3^{\frac {1}{3}}\right )}{\left (-2 \pi 3^{\frac {5}{6}}+3 \,3^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )^{2}\right ) \operatorname {AiryAi}\left (-x \right )+\operatorname {AiryBi}\left (-x \right ) \left (3 \,3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}+2 \pi 3^{\frac {1}{3}}\right )} \]
✓ Solution by Mathematica
Time used: 1.986 (sec). Leaf size: 145
DSolve[{y'[x]==y[x]^2+x,{y[0]==1}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\sqrt [3]{3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (x^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2 x^{3/2}}{3}\right )-x^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2 x^{3/2}}{3}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2 x^{3/2}}{3}\right )\right )-2 x^{3/2} \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2 x^{3/2}}{3}\right )}{2 x \left (\operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (\frac {1}{3},\frac {2 x^{3/2}}{3}\right )-\sqrt [3]{3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2 x^{3/2}}{3}\right )\right )} \]