72.3.3 problem 3
Internal
problem
ID
[14667]
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
Date
solved
:
Tuesday, January 28, 2025 at 06:46:56 AM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }&=y^{2}-4 t \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&={\frac {1}{2}} \end{align*}
✓ Solution by Maple
Time used: 0.132 (sec). Leaf size: 110
dsolve([diff(y(t),t)=y(t)^2-4*t,y(0) = 1/2],y(t), singsol=all)
\[
y = \frac {\left (\left (3 \,2^{{2}/{3}} 3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}-\pi 3^{{1}/{3}}\right ) \operatorname {AiryBi}\left (1, 2^{{2}/{3}} t \right )+\left (3^{{5}/{6}} \pi +3 \Gamma \left (\frac {2}{3}\right )^{2} 6^{{2}/{3}}\right ) \operatorname {AiryAi}\left (1, 2^{{2}/{3}} t \right )\right ) 2^{{2}/{3}}}{\left (-3^{{5}/{6}} \pi -3 \Gamma \left (\frac {2}{3}\right )^{2} 6^{{2}/{3}}\right ) \operatorname {AiryAi}\left (2^{{2}/{3}} t \right )+\operatorname {AiryBi}\left (2^{{2}/{3}} t \right ) \left (-3 \,2^{{2}/{3}} 3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}+\pi 3^{{1}/{3}}\right )}
\]
✓ Solution by Mathematica
Time used: 9.478 (sec). Leaf size: 193
DSolve[{D[y[t],t]==y[t]^2-4*t,{y[0]==1/2}},y[t],t,IncludeSingularSolutions -> True]
\[
y(t)\to -\frac {4 i t^{3/2} \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (-\frac {2}{3},\frac {4}{3} i t^{3/2}\right )+2^{2/3} \sqrt [3]{3} \left (\sqrt {3}-i\right ) \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (2 t^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {4}{3} i t^{3/2}\right )-2 t^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {4}{3} i t^{3/2}\right )-i \operatorname {BesselJ}\left (-\frac {1}{3},\frac {4}{3} i t^{3/2}\right )\right )}{2 t \left (2^{2/3} \sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {BesselJ}\left (-\frac {1}{3},\frac {4}{3} i t^{3/2}\right )+\operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (\frac {1}{3},\frac {4}{3} i t^{3/2}\right )\right )}
\]