Internal problem ID [12233]
Internal file name [OUTPUT/10886_Thursday_September_28_2023_01_08_06_AM_94705325/index.tex]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 4, Second and Higher Order Linear Differential Equations. Problems page
221
Problem number: Problem 1(m).
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.
\[ \boxed {2 y^{\prime \prime }+3 y^{\prime }+4 x^{2} y=1} \]
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: 0.109 (sec). Leaf size: 376
dsolve(2*diff(y(x),x$2)+3*diff(y(x),x)+4*x^2*y(x)=1,y(x), singsol=all)
\[ y \left (x \right ) = -48 \left (-\frac {32 \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \left (i+\frac {3 \sqrt {2}}{32}\right ) \left (\int \frac {\operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{\frac {i \sqrt {2}\, x^{2}}{2}+\frac {3 x}{4}}}{1563 i \operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )+2048 \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerU}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \left (i+\frac {3 \sqrt {2}}{32}\right )}d x \right )}{3}+\left (\int \frac {\operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{\frac {i \sqrt {2}\, x^{2}}{2}+\frac {3 x}{4}}}{\left (192 i \sqrt {2}-2048\right ) \operatorname {KummerU}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )-1563 \operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (-\frac {9 i \sqrt {2}}{128}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )}d x \right ) \left (i \sqrt {2}-\frac {32}{3}\right ) \operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )-\frac {\operatorname {KummerU}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) c_{1}}{48}-\frac {\operatorname {KummerM}\left (\frac {3}{4}-\frac {9 i \sqrt {2}}{128}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) c_{2}}{48}\right ) x \,{\mathrm e}^{-\frac {\left (i \sqrt {2}\, x +\frac {3}{2}\right ) x}{2}} \]
✓ Solution by Mathematica
Time used: 11.093 (sec). Leaf size: 553
DSolve[2*y''[x]+3*y'[x]+4*x^2*y[x]==1,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{\frac {1}{4} x \left (-3-2 i \sqrt {2} x\right )} \left (\operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} x^2\right ) \int _1^x\frac {(8+8 i) e^{\frac {1}{4} K[2] \left (2 i \sqrt {2} K[2]+3\right )} \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} K[2]\right )}{\left (9+16 i \sqrt {2}\right ) \left (\sqrt [4]{2} \operatorname {HermiteH}\left (-\frac {3}{2}+\frac {9 i}{32 \sqrt {2}},\frac {(1+i) K[2]}{\sqrt [4]{2}}\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} K[2]^2\right )+(1+i) \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\frac {(1+i) K[2]}{\sqrt [4]{2}}\right ) \operatorname {Hypergeometric1F1}\left (\frac {5}{4}-\frac {9 i}{64 \sqrt {2}},\frac {3}{2},i \sqrt {2} K[2]^2\right ) K[2]\right )}dK[2]+\operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} x\right ) \int _1^x\frac {16 e^{\frac {1}{4} K[1] \left (2 i \sqrt {2} K[1]+3\right )} \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} K[1]^2\right )}{\sqrt [4]{-2} \left (-32+9 i \sqrt {2}\right ) \operatorname {HermiteH}\left (-\frac {3}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} K[1]\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} K[1]^2\right )+2 \left (-9-16 i \sqrt {2}\right ) \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} K[1]\right ) \operatorname {Hypergeometric1F1}\left (\frac {5}{4}-\frac {9 i}{64 \sqrt {2}},\frac {3}{2},i \sqrt {2} K[1]^2\right ) K[1]}dK[1]+c_1 \operatorname {HermiteH}\left (-\frac {1}{2}+\frac {9 i}{32 \sqrt {2}},\sqrt [4]{-2} x\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {1}{4}-\frac {9 i}{64 \sqrt {2}},\frac {1}{2},i \sqrt {2} x^2\right )\right ) \]