Internal problem ID [11007]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {t^{3} y^{\prime \prime }-2 y^{\prime } t +y-t^{4}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 120
dsolve(t^3*diff(y(t),t$2)-2*t*diff(y(t),t)+y(t)=t^4,y(t), singsol=all)
\[ y \left (t \right ) = {\mathrm e}^{-\frac {1}{t}} \left (\operatorname {BesselI}\left (0, \frac {1}{t}\right )+\operatorname {BesselI}\left (1, \frac {1}{t}\right )\right ) c_{2} +{\mathrm e}^{-\frac {1}{t}} \left (-\operatorname {BesselK}\left (0, \frac {1}{t}\right )+\operatorname {BesselK}\left (1, \frac {1}{t}\right )\right ) c_{1} -\left (\left (\operatorname {BesselI}\left (0, \frac {1}{t}\right )+\operatorname {BesselI}\left (1, \frac {1}{t}\right )\right ) \left (\int t \left (-\operatorname {BesselK}\left (0, \frac {1}{t}\right )+\operatorname {BesselK}\left (1, \frac {1}{t}\right )\right ) {\mathrm e}^{\frac {1}{t}}d t \right )+\left (\int t \left (\operatorname {BesselI}\left (0, \frac {1}{t}\right )+\operatorname {BesselI}\left (1, \frac {1}{t}\right )\right ) {\mathrm e}^{\frac {1}{t}}d t \right ) \left (\operatorname {BesselK}\left (0, \frac {1}{t}\right )-\operatorname {BesselK}\left (1, \frac {1}{t}\right )\right )\right ) {\mathrm e}^{-\frac {1}{t}} \]
✓ Solution by Mathematica
Time used: 12.134 (sec). Leaf size: 272
DSolve[t^3*y''[t]-2*t*y'[t]+y[t]==t^4,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]}| {c} \frac {1}{2} \\ -1,0 \\ \\ \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]}| {c} \frac {1}{2} \\ -1,0 \\ \\ \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}| {c} \frac {1}{2} \\ -1,0 \\ \\ \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]}| {c} \frac {1}{2} \\ -1,0 \\ \\ \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*}