Internal problem ID [12931]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.4 page 61
Problem number: 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Riccati, _special]]
\[ \boxed {y^{\prime }-y^{2}=-4 t} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {1}{2}}\right ] \end {align*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 115
dsolve([diff(y(t),t)=y(t)^2-4*t,y(0) = 1/2],y(t), singsol=all)
\[ y \left (t \right ) = \frac {2^{\frac {2}{3}} \left (\left (3 \,2^{\frac {2}{3}} 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}-\pi 3^{\frac {1}{3}}\right ) \operatorname {AiryBi}\left (1, 2^{\frac {2}{3}} t \right )+\operatorname {AiryAi}\left (1, 2^{\frac {2}{3}} t \right ) \left (3 \Gamma \left (\frac {2}{3}\right )^{2} 6^{\frac {2}{3}}+3^{\frac {5}{6}} \pi \right )\right )}{\left (-3 \Gamma \left (\frac {2}{3}\right )^{2} 6^{\frac {2}{3}}-3^{\frac {5}{6}} \pi \right ) \operatorname {AiryAi}\left (2^{\frac {2}{3}} t \right )+\operatorname {AiryBi}\left (2^{\frac {2}{3}} t \right ) \left (-3 \,2^{\frac {2}{3}} 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}+\pi 3^{\frac {1}{3}}\right )} \]
✓ Solution by Mathematica
Time used: 10.151 (sec). Leaf size: 193
DSolve[{y'[t]==y[t]^2-4*t,{y[0]==1/2}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to -\frac {4 i t^{3/2} \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (-\frac {2}{3},\frac {4}{3} i t^{3/2}\right )+2^{2/3} \sqrt [3]{3} \left (\sqrt {3}-i\right ) \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (2 t^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {4}{3} i t^{3/2}\right )-2 t^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {4}{3} i t^{3/2}\right )-i \operatorname {BesselJ}\left (-\frac {1}{3},\frac {4}{3} i t^{3/2}\right )\right )}{2 t \left (2^{2/3} \sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {BesselJ}\left (-\frac {1}{3},\frac {4}{3} i t^{3/2}\right )+\operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (\frac {1}{3},\frac {4}{3} i t^{3/2}\right )\right )} \]