14.4.2 problem 2
Internal
problem
ID
[2520]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
1.
First
order
differential
equations.
Section
1.10.
Existence-uniqueness
theorem.
Excercises
page
80
Problem
number
:
2
Date
solved
:
Tuesday, January 28, 2025 at 02:36:10 PM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }&=t^{2}+y^{2} \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=1 \end{align*}
✓ Solution by Maple
Time used: 0.247 (sec). Leaf size: 139
dsolve([diff(y(t),t)=t^2+y(t)^2,y(0) = 1],y(t), singsol=all)
\[
y = \left \{\begin {array}{cc} -\frac {\left (\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {t^{2}}{2}\right ) \left (\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right )-\operatorname {BesselY}\left (-\frac {3}{4}, \frac {t^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}\right ) t}{\left (\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{2}\right )-\operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}} & t <0 \\ 1 & t =0 \\ -\frac {\left (\operatorname {BesselY}\left (-\frac {3}{4}, \frac {t^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}+\left (-\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right ) \operatorname {BesselJ}\left (-\frac {3}{4}, \frac {t^{2}}{2}\right )\right ) t}{\operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}+\left (-\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{2}\right )} & 0<t \end {array}\right .
\]
✓ Solution by Mathematica
Time used: 0.369 (sec). Leaf size: 114
DSolve[{D[y[t],t]==t^2+y[t]^2,{y[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\[
y(t)\to \frac {\operatorname {Gamma}\left (\frac {3}{4}\right ) \left (t^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {t^2}{2}\right )-t^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {t^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {t^2}{2}\right )\right )-t^2 \operatorname {Gamma}\left (\frac {1}{4}\right ) \operatorname {BesselJ}\left (-\frac {3}{4},\frac {t^2}{2}\right )}{t \left (\operatorname {Gamma}\left (\frac {1}{4}\right ) \operatorname {BesselJ}\left (\frac {1}{4},\frac {t^2}{2}\right )-2 \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {BesselJ}\left (-\frac {1}{4},\frac {t^2}{2}\right )\right )}
\]