72.3.3 problem 3
Internal
problem
ID
[14596]
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
:
Monday, March 31, 2025 at 12:39:36 PM
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*}
✓ Maple. Time used: 0.213 (sec). Leaf size: 125
ode:=diff(y(t),t) = y(t)^2-4*t;
ic:=y(0) = 1/2;
dsolve([ode,ic],y(t), singsol=all);
\[
y = \frac {\left (-3 \,2^{{2}/{3}} \left (\operatorname {AiryAi}\left (1, 2^{{2}/{3}} t \right ) 3^{{2}/{3}}+\operatorname {AiryBi}\left (1, 2^{{2}/{3}} t \right ) 3^{{1}/{6}}\right ) \Gamma \left (\frac {2}{3}\right )^{2}-\operatorname {AiryAi}\left (1, 2^{{2}/{3}} t \right ) 3^{{5}/{6}} \pi +\operatorname {AiryBi}\left (1, 2^{{2}/{3}} t \right ) \pi 3^{{1}/{3}}\right ) 2^{{2}/{3}}}{3^{{5}/{6}} \pi \operatorname {AiryAi}\left (2^{{2}/{3}} t \right )+3 \,3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2} 2^{{2}/{3}} \operatorname {AiryBi}\left (2^{{2}/{3}} t \right )+3 \,6^{{2}/{3}} \Gamma \left (\frac {2}{3}\right )^{2} \operatorname {AiryAi}\left (2^{{2}/{3}} t \right )-\pi 3^{{1}/{3}} \operatorname {AiryBi}\left (2^{{2}/{3}} t \right )}
\]
✓ Mathematica. Time used: 9.478 (sec). Leaf size: 193
ode=D[y[t],t]==y[t]^2-4*t;
ic={y[0]==1/2};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to -\frac {4 i t^{3/2} \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (-\frac {2}{3},\frac {4}{3} i t^{3/2}\right )+2^{2/3} \sqrt [3]{3} \left (\sqrt {3}-i\right ) \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (2 t^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {4}{3} i t^{3/2}\right )-2 t^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {4}{3} i t^{3/2}\right )-i \operatorname {BesselJ}\left (-\frac {1}{3},\frac {4}{3} i t^{3/2}\right )\right )}{2 t \left (2^{2/3} \sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {BesselJ}\left (-\frac {1}{3},\frac {4}{3} i t^{3/2}\right )+\operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (\frac {1}{3},\frac {4}{3} i t^{3/2}\right )\right )}
\]
✗ Sympy
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(4*t - y(t)**2 + Derivative(y(t), t),0)
ics = {y(0): 1/2}
dsolve(ode,func=y(t),ics=ics)
TypeError : bad operand type for unary -: list