61.5.5 problem Problem 1(e)
Internal
problem
ID
[15372]
Book
:
APPLIED
DIFFERENTIAL
EQUATIONS
The
Primary
Course
by
Vladimir
A.
Dobrushkin.
CRC
Press
2015
Section
:
Chapter
6.
Introduction
to
Systems
of
ODEs.
Problems
page
408
Problem
number
:
Problem
1(e)
Date
solved
:
Friday, October 03, 2025 at 07:30:20 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} t^{3} y^{\prime \prime }-2 t y^{\prime }+y&=t^{4} \end{align*}
✓ Maple. Time used: 0.035 (sec). Leaf size: 114
ode:=t^3*diff(diff(y(t),t),t)-2*t*diff(y(t),t)+y(t) = t^4;
dsolve(ode,y(t), singsol=all);
\[
y = -{\mathrm e}^{-\frac {1}{t}} \left (\left (-\operatorname {BesselI}\left (0, \frac {1}{t}\right )-\operatorname {BesselI}\left (1, \frac {1}{t}\right )\right ) \int {\mathrm e}^{\frac {1}{t}} \left (\operatorname {BesselK}\left (0, \frac {1}{t}\right )-\operatorname {BesselK}\left (1, \frac {1}{t}\right )\right ) t d t +\int {\mathrm e}^{\frac {1}{t}} \left (\operatorname {BesselI}\left (0, \frac {1}{t}\right )+\operatorname {BesselI}\left (1, \frac {1}{t}\right )\right ) t d t \left (\operatorname {BesselK}\left (0, \frac {1}{t}\right )-\operatorname {BesselK}\left (1, \frac {1}{t}\right )\right )-c_1 \operatorname {BesselK}\left (0, \frac {1}{t}\right )+c_1 \operatorname {BesselK}\left (1, \frac {1}{t}\right )-c_2 \operatorname {BesselI}\left (0, \frac {1}{t}\right )-c_2 \operatorname {BesselI}\left (1, \frac {1}{t}\right )\right )
\]
✓ Mathematica. Time used: 20.997 (sec). Leaf size: 272
ode=t^3*D[y[t],{t,2}]-2*t*D[y[t],t]+y[t]==t^4;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to e^{-1/t} \left (\operatorname {BesselI}\left (0,\frac {1}{t}\right )+\operatorname {BesselI}\left (1,\frac {1}{t}\right )\right ) \left (\int _1^t\frac {2 e^{\frac {2}{K[1]}} \sqrt {\pi } K[1]^3 G_{1,2}^{2,0}\left (\frac {2}{K[1]}| \begin {array}{c} \frac {1}{2} \\ -1,0 \\ \end {array} \right )}{e^{\frac {1}{K[1]}} \sqrt {\pi } \left (\operatorname {BesselI}\left (0,\frac {1}{K[1]}\right )-\operatorname {BesselI}\left (2,\frac {1}{K[1]}\right )\right ) G_{1,2}^{2,0}\left (\frac {2}{K[1]}| \begin {array}{c} \frac {1}{2} \\ -1,0 \\ \end {array} \right )-2 \left (\operatorname {BesselI}\left (0,\frac {1}{K[1]}\right )+\operatorname {BesselI}\left (1,\frac {1}{K[1]}\right )\right ) K_1\left (\frac {1}{K[1]}\right ) K[1]}dK[1]+c_1\right )+G_{1,2}^{2,0}\left (\frac {2}{t}| \begin {array}{c} \frac {1}{2} \\ -1,0 \\ \end {array} \right ) \left (\int _1^t-\frac {2 e^{\frac {1}{K[2]}} \sqrt {\pi } \left (\operatorname {BesselI}\left (0,\frac {1}{K[2]}\right )+\operatorname {BesselI}\left (1,\frac {1}{K[2]}\right )\right ) K[2]^3}{e^{\frac {1}{K[2]}} \sqrt {\pi } \left (\operatorname {BesselI}\left (0,\frac {1}{K[2]}\right )-\operatorname {BesselI}\left (2,\frac {1}{K[2]}\right )\right ) G_{1,2}^{2,0}\left (\frac {2}{K[2]}| \begin {array}{c} \frac {1}{2} \\ -1,0 \\ \end {array} \right )-2 \left (\operatorname {BesselI}\left (0,\frac {1}{K[2]}\right )+\operatorname {BesselI}\left (1,\frac {1}{K[2]}\right )\right ) K_1\left (\frac {1}{K[2]}\right ) K[2]}dK[2]+c_2\right ) \end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-t**4 + t**3*Derivative(y(t), (t, 2)) - 2*t*Derivative(y(t), t) + y(t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
NotImplementedError : The given ODE Derivative(y(t), t) - (t**3*(-t + Derivative(y(t), (t, 2))) + y(t))/(2*t) cannot be solved by the factorable group method