13.5.13 problem 16

Internal problem ID [2359]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.10. Page 80
Problem number : 16
Date solved : Monday, January 27, 2025 at 05:50:05 AM
CAS classification : [[_Riccati, _special]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.190 (sec). Leaf size: 42

dsolve([diff(y(t),t)=t^2+y(t)^2,y(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.392 (sec). Leaf size: 68

DSolve[{D[y[t],t]==t^2+y[t]^2,{y[0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -\frac {t^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {t^2}{2}\right )-t^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {t^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {t^2}{2}\right )}{2 t \operatorname {BesselJ}\left (-\frac {1}{4},\frac {t^2}{2}\right )} \]