80.7.34 problem D 3
Internal
problem
ID
[21353]
Book
:
A
Textbook
on
Ordinary
Differential
Equations
by
Shair
Ahmad
and
Antonio
Ambrosetti.
Second
edition.
ISBN
978-3-319-16407-6.
Springer
2015
Section
:
Chapter
7.
System
of
first
order
equations.
Excercise
7.6
at
page
162
Problem
number
:
D
3
Date
solved
:
Sunday, October 12, 2025 at 05:51:27 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )-t y \left (t \right )&=1\\ \frac {d}{d t}y \left (t \right )-t \left (\frac {d}{d t}x \left (t \right )\right )&=3 \end{align*}
✓ Maple. Time used: 0.106 (sec). Leaf size: 274
ode:=[diff(x(t),t)-t*y(t) = 1, diff(y(t),t)-t*diff(x(t),t) = 3];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \int \frac {5 \,{\mathrm e}^{\frac {t^{3}}{6}} \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {t^{3}}{3}\right ) \left (t^{3}\right )^{{1}/{3}} 243^{{5}/{6}} t^{3}+18 \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {5}{6}, \frac {t^{3}}{3}\right ) \left (t^{3}\right )^{{1}/{6}} {\mathrm e}^{\frac {t^{3}}{6}} 9^{{2}/{3}} t^{4}+20 \operatorname {WhittakerM}\left (\frac {7}{6}, \frac {2}{3}, \frac {t^{3}}{3}\right ) {\mathrm e}^{\frac {t^{3}}{6}} \left (t^{3}\right )^{{1}/{3}} 243^{{5}/{6}}+90 \left (t^{3}\right )^{{1}/{6}} \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {t^{3}}{3}\right ) {\mathrm e}^{\frac {t^{3}}{6}} 9^{{2}/{3}} t +180 \,{\mathrm e}^{\frac {t^{3}}{3}} c_2 \sqrt {t^{3}}\, t^{2}}{180 t \sqrt {t^{3}}}d t +t +c_1 \\
y \left (t \right ) &= \frac {5 \,{\mathrm e}^{\frac {t^{3}}{6}} \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {t^{3}}{3}\right ) \left (t^{3}\right )^{{1}/{3}} 243^{{5}/{6}} t^{3}+18 \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {5}{6}, \frac {t^{3}}{3}\right ) \left (t^{3}\right )^{{1}/{6}} {\mathrm e}^{\frac {t^{3}}{6}} 9^{{2}/{3}} t^{4}+20 \operatorname {WhittakerM}\left (\frac {7}{6}, \frac {2}{3}, \frac {t^{3}}{3}\right ) {\mathrm e}^{\frac {t^{3}}{6}} \left (t^{3}\right )^{{1}/{3}} 243^{{5}/{6}}+90 \left (t^{3}\right )^{{1}/{6}} \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {t^{3}}{3}\right ) {\mathrm e}^{\frac {t^{3}}{6}} 9^{{2}/{3}} t +180 \,{\mathrm e}^{\frac {t^{3}}{3}} c_2 \sqrt {t^{3}}\, t^{2}}{180 \sqrt {t^{3}}\, t^{2}} \\
\end{align*}
✓ Mathematica. Time used: 0.902 (sec). Leaf size: 242
ode={D[x[t],t]-t*y[t]==1,D[y[t],t]-t*D[x[t],t]==3};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \int _1^t\left (\frac {e^{-\frac {1}{3} K[1]^3} \Gamma \left (\frac {2}{3},-\frac {1}{3} K[1]^3\right ) (K[1]+3) K[1]^2}{\sqrt [3]{3} \left (-K[1]^3\right )^{2/3}}+1\right )dK[1]+\frac {\Gamma \left (\frac {2}{3},-\frac {t^3}{3}\right ) \left (-3 \sqrt [3]{-t^6} t \Gamma \left (\frac {1}{3},\frac {t^3}{3}\right )+3^{2/3} c_1 \sqrt [3]{t^3} \sqrt [3]{-t^6}+\frac {\sqrt [3]{3} t^5 \Gamma \left (\frac {2}{3},\frac {t^3}{3}\right )}{\left (-t^3\right )^{2/3}}\right )}{3 t \left (t^3\right )^{2/3}}+c_2\\ y(t)&\to -\frac {e^{\frac {t^3}{3}} \left (3 \sqrt [3]{3} \left (t^3\right )^{2/3} \Gamma \left (\frac {1}{3},\frac {t^3}{3}\right )+t \left (3^{2/3} \sqrt [3]{t^3} \Gamma \left (\frac {2}{3},\frac {t^3}{3}\right )-3 c_1 t\right )\right )}{3 t^2} \end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-t*y(t) + Derivative(x(t), t) - 1,0),Eq(-t*Derivative(x(t), t) + Derivative(y(t), t) - 3,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
Timed Out