Internal problem ID [6545]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Riccati, _special]]
\[ \boxed {-y^{2}+y^{\prime }=-x} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.171 (sec). Leaf size: 90
dsolve([diff(y(x),x)=y(x)^2-x,y(0) = 1],y(x), singsol=all)
\[ y \left (x \right ) = \frac {-2 \operatorname {AiryAi}\left (1, x\right ) 3^{\frac {5}{6}} \pi -3 \operatorname {AiryAi}\left (1, x\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}}-3 \operatorname {AiryBi}\left (1, x\right ) 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}+2 \operatorname {AiryBi}\left (1, x\right ) 3^{\frac {1}{3}} \pi }{2 \operatorname {AiryAi}\left (x \right ) 3^{\frac {5}{6}} \pi +3 \operatorname {AiryAi}\left (x \right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}}+3 \operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}-2 \operatorname {AiryBi}\left (x \right ) 3^{\frac {1}{3}} \pi } \]
✓ Solution by Mathematica
Time used: 7.282 (sec). Leaf size: 164
DSolve[{y'[x]==y[x]^2-x,{y[0]==1}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\sqrt [3]{-3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (i x^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} i x^{3/2}\right )-i x^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} i x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )\right )-2 i x^{3/2} \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} i x^{3/2}\right )}{2 x \left (\operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} i x^{3/2}\right )-\sqrt [3]{-3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )\right )} \]