5.1.8 problem 17
Internal
problem
ID
[1469]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
11th
ed.,
Boyce,
DiPrima,
Meade
Section
:
Chapter
4.1,
Higher
order
linear
differential
equations.
General
theory.
page
173
Problem
number
:
17
Date
solved
:
Friday, October 03, 2025 at 01:20:27 AM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} t y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }+t y&=0 \end{align*}
✓ Maple. Time used: 0.095 (sec). Leaf size: 144
ode:=t*diff(diff(diff(y(t),t),t),t)+2*diff(diff(y(t),t),t)-diff(y(t),t)+t*y(t) = 0;
dsolve(ode,y(t), singsol=all);
\[
y = {\mathrm e}^{-\frac {t \left (i \sqrt {3}-1\right )}{2}} \left (\int \operatorname {KummerU}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) {\mathrm e}^{-\frac {t \left (i \sqrt {3}+3\right )}{2}}d t \operatorname {KummerM}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) c_3 -\int \operatorname {KummerM}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) {\mathrm e}^{-\frac {t \left (i \sqrt {3}+3\right )}{2}}d t \operatorname {KummerU}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) c_3 +c_1 \operatorname {KummerM}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right )+c_2 \operatorname {KummerU}\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right )\right )
\]
✓ Mathematica. Time used: 0.383 (sec). Leaf size: 520
ode=t*D[ y[t],{t,3}]+2*D[y[t],{t,2}]-D[y[t],t]+t*y[t]==0;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to e^{\frac {1}{2} \left (t-i \sqrt {3} t\right )} \left (c_3 \operatorname {HypergeometricU}\left (\frac {1}{6} \left (3-i \sqrt {3}\right ),1,i \sqrt {3} t\right ) \int _1^t\frac {2 e^{\frac {1}{2} i \left (3 i+\sqrt {3}\right ) K[1]} \operatorname {LaguerreL}\left (\frac {1}{6} i \left (3 i+\sqrt {3}\right ),i \sqrt {3} K[1]\right )}{\left (-1-i \sqrt {3}\right ) K[1] \left (\operatorname {Hypergeometric1F1}\left (\frac {1}{6} \left (9-i \sqrt {3}\right ),2,i \sqrt {3} K[1]\right ) \operatorname {HypergeometricU}\left (\frac {1}{6} \left (3-i \sqrt {3}\right ),1,i \sqrt {3} K[1]\right )+\operatorname {HypergeometricU}\left (\frac {1}{6} \left (9-i \sqrt {3}\right ),2,i \sqrt {3} K[1]\right ) \operatorname {LaguerreL}\left (\frac {1}{6} i \left (3 i+\sqrt {3}\right ),i \sqrt {3} K[1]\right )\right )}dK[1]+c_3 \operatorname {LaguerreL}\left (\frac {1}{6} i \left (3 i+\sqrt {3}\right ),i \sqrt {3} t\right ) \int _1^t-\frac {2 i e^{\frac {1}{2} i \left (3 i+\sqrt {3}\right ) K[2]} \operatorname {HypergeometricU}\left (\frac {1}{6} \left (3-i \sqrt {3}\right ),1,i \sqrt {3} K[2]\right )}{\left (-i+\sqrt {3}\right ) K[2] \left (\operatorname {Hypergeometric1F1}\left (\frac {1}{6} \left (9-i \sqrt {3}\right ),2,i \sqrt {3} K[2]\right ) \operatorname {HypergeometricU}\left (\frac {1}{6} \left (3-i \sqrt {3}\right ),1,i \sqrt {3} K[2]\right )+\operatorname {HypergeometricU}\left (\frac {1}{6} \left (9-i \sqrt {3}\right ),2,i \sqrt {3} K[2]\right ) \operatorname {LaguerreL}\left (\frac {1}{6} i \left (3 i+\sqrt {3}\right ),i \sqrt {3} K[2]\right )\right )}dK[2]+c_1 \operatorname {HypergeometricU}\left (\frac {1}{6} \left (3-i \sqrt {3}\right ),1,i \sqrt {3} t\right )+c_2 \operatorname {LaguerreL}\left (\frac {1}{6} i \left (3 i+\sqrt {3}\right ),i \sqrt {3} t\right )\right ) \end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(t*y(t) + t*Derivative(y(t), (t, 3)) - Derivative(y(t), t) + 2*Derivative(y(t), (t, 2)),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
NotImplementedError : The given ODE -t*y(t) - t*Derivative(y(t), (t, 3)) + Derivative(y(t), t) - 2*Derivative(y(t), (t, 2)) cannot be solved by the factorable group method