56.2.37 problem 36
Internal
problem
ID
[8841]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
36
Date
solved
:
Sunday, March 30, 2025 at 01:41:47 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }-x^{2} y-x^{4}&=0 \end{align*}
✓ Maple. Time used: 0.075 (sec). Leaf size: 124
ode:=diff(diff(y(x),x),x)-x^2*y(x)-x^4 = 0;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\sqrt {x}\, \left (-\frac {6 x^{5} \operatorname {csgn}\left (x \right ) \pi ^{2} \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {5}{4}\right ], \left [\frac {3}{4}, \frac {5}{2}\right ], \frac {x^{4}}{16}\right )}{5}+\left (2 x^{6} \Gamma \left (\frac {3}{4}\right ) \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}\right ], \left [\frac {5}{4}, \frac {5}{2}\right ], \frac {x^{4}}{16}\right )+\pi \left (x^{6} \Gamma \left (\frac {3}{4}\right ) \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}\right ], \left [\frac {19}{8}, \frac {5}{2}\right ], \frac {x^{4}}{16}\right ) \sqrt {2}-12 \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_2 -12 \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_1 \right )\right ) \Gamma \left (\frac {3}{4}\right )\right )}{12 \Gamma \left (\frac {3}{4}\right ) \pi }
\]
✓ Mathematica. Time used: 3.328 (sec). Leaf size: 213
ode=D[y[x],{x,2}]-x^2*y[x]-x^4==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt {2} x\right ) \left (\int _1^x\frac {K[1]^4 \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},i \sqrt {2} K[1]\right )}{\sqrt {2} \left (\operatorname {HermiteH}\left (-\frac {1}{2},K[1]\right ) \left (i \operatorname {HermiteH}\left (\frac {1}{2},i K[1]\right )+2 \operatorname {HermiteH}\left (-\frac {1}{2},i K[1]\right ) K[1]\right )-\operatorname {HermiteH}\left (-\frac {1}{2},i K[1]\right ) \operatorname {HermiteH}\left (\frac {1}{2},K[1]\right )\right )}dK[1]+c_1\right )+\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},i \sqrt {2} x\right ) \left (\int _1^x\frac {K[2]^4 \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt {2} K[2]\right )}{\sqrt {2} \left (\operatorname {HermiteH}\left (-\frac {1}{2},i K[2]\right ) \operatorname {HermiteH}\left (\frac {1}{2},K[2]\right )+\operatorname {HermiteH}\left (-\frac {1}{2},K[2]\right ) \left (-i \operatorname {HermiteH}\left (\frac {1}{2},i K[2]\right )-2 \operatorname {HermiteH}\left (-\frac {1}{2},i K[2]\right ) K[2]\right )\right )}dK[2]+c_2\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**4 - x**2*y(x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve -x**4 - x**2*y(x) + Derivative(y(x), (x, 2))