74.3.1 problem 1 (a)

Internal problem ID [15861]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.1, page 32
Problem number : 1 (a)
Date solved : Tuesday, January 28, 2025 at 08:10:31 AM
CAS classification : [_Riccati]

\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: 1.010 (sec). Leaf size: 59

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

\[ y = -\left (\left \{\begin {array}{cc} 0 & t =0 \\ \frac {\left (\operatorname {BesselI}\left (-\frac {3}{4}, \frac {t^{2}}{2}\right ) \pi \sqrt {2}-2 \operatorname {BesselK}\left (\frac {3}{4}, \frac {t^{2}}{2}\right )\right ) t}{\pi \sqrt {2}\, \operatorname {BesselI}\left (\frac {1}{4}, \frac {t^{2}}{2}\right )+2 \operatorname {BesselK}\left (\frac {1}{4}, \frac {t^{2}}{2}\right )} & \operatorname {otherwise} \end {array}\right .\right ) \]

✓ Solution by Mathematica

Time used: 0.613 (sec). Leaf size: 81

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

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