72.7.18 problem 18
Internal
problem
ID
[14680]
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
1.9
page
133
Problem
number
:
18
Date
solved
:
Thursday, March 13, 2025 at 04:13:53 AM
CAS
classification
:
[_linear]
\begin{align*} y^{\prime }&=\frac {y}{\sqrt {t^{3}-3}}+t \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 33
ode:=diff(y(t),t) = y(t)/(t^3-3)^(1/2)+t;
dsolve(ode,y(t), singsol=all);
\[
y = \left (\int t \,{\mathrm e}^{-\int \frac {1}{\sqrt {t^{3}-3}}d t}d t +c_{1} \right ) {\mathrm e}^{\int \frac {1}{\sqrt {t^{3}-3}}d t}
\]
✓ Mathematica. Time used: 0.212 (sec). Leaf size: 110
ode=D[y[t],t]==y[t]/Sqrt[t^3-3]+t;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to e^{\frac {t \sqrt {1-\frac {t^3}{3}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {t^3}{3}\right )}{\sqrt {t^3-3}}} \left (\int _1^t\exp \left (-\frac {\operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {K[1]^3}{3}\right ) K[1] \sqrt {1-\frac {K[1]^3}{3}}}{\sqrt {K[1]^3-3}}\right ) K[1]dK[1]+c_1\right )
\]
✓ Sympy. Time used: 5.336 (sec). Leaf size: 136
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-t + Derivative(y(t), t) - y(t)/sqrt(t**3 - 3),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \frac {C_{1} + \int t e^{\frac {\sqrt {3} i t {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {t^{3}}{3}} \right )}}{3}}\, dt + \int \frac {y{\left (t \right )} e^{\frac {\sqrt {3} i t \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {t^{3}}{3}} \right )}}{9 \Gamma \left (\frac {4}{3}\right )}}}{\sqrt {t^{3} - 3}}\, dt}{e^{\frac {\sqrt {3} i t {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {t^{3}}{3}} \right )}}{3}} + \int \frac {e^{\frac {\sqrt {3} i t {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {t^{3}}{3}} \right )}}{3}}}{\sqrt {t^{3} - 3}}\, dt}
\]