67.5.5 problem Problem 1(e)
Internal
problem
ID
[14091]
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
:
Tuesday, January 28, 2025 at 08:25:27 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} t^{3} y^{\prime \prime }-2 y^{\prime } t +y&=t^{4} \end{align*}
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 110
dsolve(t^3*diff(y(t),t$2)-2*t*diff(y(t),t)+y(t)=t^4,y(t), singsol=all)
\[
y = -\left (\left (\operatorname {BesselI}\left (0, \frac {1}{t}\right )+\operatorname {BesselI}\left (1, \frac {1}{t}\right )\right ) \left (\int t \left (-\operatorname {BesselK}\left (0, \frac {1}{t}\right )+\operatorname {BesselK}\left (1, \frac {1}{t}\right )\right ) {\mathrm e}^{\frac {1}{t}}d t \right )+\left (\int t \left (\operatorname {BesselI}\left (0, \frac {1}{t}\right )+\operatorname {BesselI}\left (1, \frac {1}{t}\right )\right ) {\mathrm e}^{\frac {1}{t}}d t \right ) \left (\operatorname {BesselK}\left (0, \frac {1}{t}\right )-\operatorname {BesselK}\left (1, \frac {1}{t}\right )\right )+\operatorname {BesselK}\left (0, \frac {1}{t}\right ) c_{1} -\operatorname {BesselK}\left (1, \frac {1}{t}\right ) c_{1} -\operatorname {BesselI}\left (0, \frac {1}{t}\right ) c_{2} -\operatorname {BesselI}\left (1, \frac {1}{t}\right ) c_{2} \right ) {\mathrm e}^{-\frac {1}{t}}
\]
✓ Solution by Mathematica
Time used: 21.750 (sec). Leaf size: 272
DSolve[t^3*D[y[t],{t,2}]-2*t*D[y[t],t]+y[t]==t^4,y[t],t,IncludeSingularSolutions -> True]
\[
y(t)\to e^{-1/t} \left (\operatorname {BesselI}\left (0,\frac {1}{t}\right )+\operatorname {BesselI}\left (1,\frac {1}{t}\right )\right ) \left (\int _1^t\frac {2 e^{\frac {2}{K[1]}} \sqrt {\pi } K[1]^3 G_{1,2}^{2,0}\left (\frac {2}{K[1]}| \begin {array}{c} \frac {1}{2} \\ -1,0 \\ \end {array} \right )}{e^{\frac {1}{K[1]}} \sqrt {\pi } \left (\operatorname {BesselI}\left (0,\frac {1}{K[1]}\right )-\operatorname {BesselI}\left (2,\frac {1}{K[1]}\right )\right ) G_{1,2}^{2,0}\left (\frac {2}{K[1]}| \begin {array}{c} \frac {1}{2} \\ -1,0 \\ \end {array} \right )-2 \left (\operatorname {BesselI}\left (0,\frac {1}{K[1]}\right )+\operatorname {BesselI}\left (1,\frac {1}{K[1]}\right )\right ) K_1\left (\frac {1}{K[1]}\right ) K[1]}dK[1]+c_1\right )+G_{1,2}^{2,0}\left (\frac {2}{t}| \begin {array}{c} \frac {1}{2} \\ -1,0 \\ \end {array} \right ) \left (\int _1^t-\frac {2 e^{\frac {1}{K[2]}} \sqrt {\pi } \left (\operatorname {BesselI}\left (0,\frac {1}{K[2]}\right )+\operatorname {BesselI}\left (1,\frac {1}{K[2]}\right )\right ) K[2]^3}{e^{\frac {1}{K[2]}} \sqrt {\pi } \left (\operatorname {BesselI}\left (0,\frac {1}{K[2]}\right )-\operatorname {BesselI}\left (2,\frac {1}{K[2]}\right )\right ) G_{1,2}^{2,0}\left (\frac {2}{K[2]}| \begin {array}{c} \frac {1}{2} \\ -1,0 \\ \end {array} \right )-2 \left (\operatorname {BesselI}\left (0,\frac {1}{K[2]}\right )+\operatorname {BesselI}\left (1,\frac {1}{K[2]}\right )\right ) K_1\left (\frac {1}{K[2]}\right ) K[2]}dK[2]+c_2\right )
\]