Internal problem ID [12352]
Internal file name [OUTPUT/11005_Monday_October_02_2023_02_47_52_AM_16697868/index.tex]
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(c).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+3 y^{\prime }+\frac {y}{t}=t} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Kummer successful <- special function solution successful <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 36
dsolve(diff(y(t),t$2)+3*diff(y(t),t)+y(t)/t=t,y(t), singsol=all)
\[ y \left (t \right ) = \frac {\left (7 \,{\mathrm e}^{-3 t} \operatorname {KummerU}\left (\frac {2}{3}, 2, 3 t \right ) c_{1} +7 \,{\mathrm e}^{-3 t} \operatorname {KummerM}\left (\frac {2}{3}, 2, 3 t \right ) c_{2} +t -\frac {1}{2}\right ) t}{7} \]
✓ Solution by Mathematica
Time used: 23.552 (sec). Leaf size: 253
DSolve[y''[t]+3*y'[t]+y[t]/t==t,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to G_{1,2}^{2,0}\left (3 t\left | \begin {array}{c} \frac {2}{3} \\ 0,1 \\ \end {array} \right .\right ) \left (\int _1^t-\frac {3 \operatorname {Hypergeometric1F1}\left (\frac {4}{3},2,-3 K[2]\right ) K[2]^2}{3 \operatorname {Hypergeometric1F1}\left (\frac {4}{3},2,-3 K[2]\right ) G_{1,2}^{2,0}\left (3 K[2]\left | \begin {array}{c} \frac {2}{3} \\ 0,1 \\ \end {array} \right .\right )+3 \operatorname {Hypergeometric1F1}\left (\frac {4}{3},2,-3 K[2]\right ) G_{1,2}^{2,0}\left (3 K[2]\left | \begin {array}{c} \frac {2}{3} \\ 1,1 \\ \end {array} \right .\right )-2 \operatorname {Hypergeometric1F1}\left (\frac {7}{3},3,-3 K[2]\right ) G_{1,2}^{2,0}\left (3 K[2]\left | \begin {array}{c} \frac {5}{3} \\ 1,2 \\ \end {array} \right .\right )}dK[2]+c_2\right )-3 t \operatorname {Hypergeometric1F1}\left (\frac {4}{3},2,-3 t\right ) \left (\int _1^t\frac {G_{1,2}^{2,0}\left (3 K[1]\left | \begin {array}{c} \frac {5}{3} \\ 1,2 \\ \end {array} \right .\right )}{-9 \operatorname {Hypergeometric1F1}\left (\frac {4}{3},2,-3 K[1]\right ) G_{1,2}^{2,0}\left (3 K[1]\left | \begin {array}{c} \frac {2}{3} \\ 0,1 \\ \end {array} \right .\right )-9 \operatorname {Hypergeometric1F1}\left (\frac {4}{3},2,-3 K[1]\right ) G_{1,2}^{2,0}\left (3 K[1]\left | \begin {array}{c} \frac {2}{3} \\ 1,1 \\ \end {array} \right .\right )+6 \operatorname {Hypergeometric1F1}\left (\frac {7}{3},3,-3 K[1]\right ) G_{1,2}^{2,0}\left (3 K[1]\left | \begin {array}{c} \frac {5}{3} \\ 1,2 \\ \end {array} \right .\right )}dK[1]+c_1\right ) \]