72.3.2 problem 2
Internal
problem
ID
[14666]
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
:
2
Date
solved
:
Tuesday, January 28, 2025 at 06:46:53 AM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }&=t -y^{2} \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=1 \end{align*}
✓ Solution by Maple
Time used: 0.129 (sec). Leaf size: 89
dsolve([diff(y(t),t)=t-y(t)^2,y(0) = 1],y(t), singsol=all)
\[
y = \frac {2 \pi \operatorname {AiryAi}\left (1, t\right ) 3^{{5}/{6}}-3 \operatorname {AiryAi}\left (1, t\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}}-3 \operatorname {AiryBi}\left (1, t\right ) 3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}-2 \pi \operatorname {AiryBi}\left (1, t\right ) 3^{{1}/{3}}}{2 \pi \operatorname {AiryAi}\left (t \right ) 3^{{5}/{6}}-3 \operatorname {AiryAi}\left (t \right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}}-3 \operatorname {AiryBi}\left (t \right ) 3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}-2 \pi \operatorname {AiryBi}\left (t \right ) 3^{{1}/{3}}}
\]
✓ Solution by Mathematica
Time used: 7.563 (sec). Leaf size: 163
DSolve[{D[y[t],t]==t-y[t]^2,{y[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\[
y(t)\to \frac {2 i t^{3/2} \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} i t^{3/2}\right )+\sqrt [3]{-3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (i t^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} i t^{3/2}\right )-i t^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} i t^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i t^{3/2}\right )\right )}{2 t \left (\sqrt [3]{-3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i t^{3/2}\right )+\operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} i t^{3/2}\right )\right )}
\]