70.12.3 problem 3
Internal
problem
ID
[18846]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
4.
Second
order
linear
equations.
Section
4.2
(Theory
of
second
order
linear
homogeneous
equations).
Problems
at
page
226
Problem
number
:
3
Date
solved
:
Friday, October 03, 2025 at 07:33:27 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} t \left (-4+t \right ) y^{\prime \prime }+3 t y^{\prime }+4 y&=2 \end{align*}
With initial conditions
\begin{align*}
y \left (3\right )&=0 \\
y^{\prime }\left (3\right )&=-1 \\
\end{align*}
✓ Maple. Time used: 0.648 (sec). Leaf size: 479
ode:=t*(t-4)*diff(diff(y(t),t),t)+3*t*diff(y(t),t)+4*y(t) = 2;
ic:=[y(3) = 0, D(y)(3) = -1];
dsolve([ode,op(ic)],y(t), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.867 (sec). Leaf size: 539
ode=t*(t-4)*D[y[t],{t,2}]+3*t*D[y[t],t]+4*y[t]==2;
ic={y[3]==0,Derivative[1][y][3]==-1};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to \frac {-2 t \operatorname {Hypergeometric2F1}\left (2-i \sqrt {3},2+i \sqrt {3},2,\frac {t}{4}\right ) \left (8 G_{2,2}^{2,0}\left (\frac {3}{4}| \begin {array}{c} -i \sqrt {3},i \sqrt {3} \\ 0,1 \\ \end {array} \right )+G_{2,2}^{2,0}\left (\frac {3}{4}| \begin {array}{c} -1-i \sqrt {3},-1+i \sqrt {3} \\ 0,0 \\ \end {array} \right )\right )+\left (40 \operatorname {Hypergeometric2F1}\left (2-i \sqrt {3},2+i \sqrt {3},2,\frac {3}{4}\right )-21 \operatorname {Hypergeometric2F1}\left (3-i \sqrt {3},3+i \sqrt {3},3,\frac {3}{4}\right )\right ) G_{2,2}^{2,0}\left (\frac {t}{4}| \begin {array}{c} -i \sqrt {3},i \sqrt {3} \\ 0,1 \\ \end {array} \right )+6 \operatorname {Hypergeometric2F1}\left (2-i \sqrt {3},2+i \sqrt {3},2,\frac {3}{4}\right ) G_{2,2}^{2,0}\left (\frac {3}{4}| \begin {array}{c} -1-i \sqrt {3},-1+i \sqrt {3} \\ 0,0 \\ \end {array} \right )+21 \operatorname {Hypergeometric2F1}\left (3-i \sqrt {3},3+i \sqrt {3},3,\frac {3}{4}\right ) G_{2,2}^{2,0}\left (\frac {3}{4}| \begin {array}{c} -i \sqrt {3},i \sqrt {3} \\ 0,1 \\ \end {array} \right )+8 \operatorname {Hypergeometric2F1}\left (2-i \sqrt {3},2+i \sqrt {3},2,\frac {3}{4}\right ) G_{2,2}^{2,0}\left (\frac {3}{4}| \begin {array}{c} -i \sqrt {3},i \sqrt {3} \\ 0,1 \\ \end {array} \right )}{42 \operatorname {Hypergeometric2F1}\left (3-i \sqrt {3},3+i \sqrt {3},3,\frac {3}{4}\right ) G_{2,2}^{2,0}\left (\frac {3}{4}| \begin {array}{c} -i \sqrt {3},i \sqrt {3} \\ 0,1 \\ \end {array} \right )+4 \operatorname {Hypergeometric2F1}\left (2-i \sqrt {3},2+i \sqrt {3},2,\frac {3}{4}\right ) \left (4 G_{2,2}^{2,0}\left (\frac {3}{4}| \begin {array}{c} -i \sqrt {3},i \sqrt {3} \\ 0,1 \\ \end {array} \right )+3 G_{2,2}^{2,0}\left (\frac {3}{4}| \begin {array}{c} -1-i \sqrt {3},-1+i \sqrt {3} \\ 0,0 \\ \end {array} \right )\right )} \end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(t*(t - 4)*Derivative(y(t), (t, 2)) + 3*t*Derivative(y(t), t) + 4*y(t) - 2,0)
ics = {y(3): 0, Subs(Derivative(y(t), t), t, 3): -1}
dsolve(ode,func=y(t),ics=ics)
NotImplementedError : The given ODE Derivative(y(t), t) - (t*(4 - t)*Derivative(y(t), (t, 2)) - 4*y(t) + 2)/(3*t) cannot be solved by the factorable group method