50.2.38 problem 37

Internal problem ID [10128]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 37
Date solved : Tuesday, September 30, 2025 at 07:03:40 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-x^{2} y-x^{4}+2&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 34

ode:=diff(diff(y(x),x),x)-x^2*y(x)-x^4+2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_2 +\sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_1 -x^{2} \]

✓ Mathematica. Time used: 2.579 (sec). Leaf size: 217

ode=D[y[x],{x,2}]-x^2*y[x]-x^4+2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt {2} x\right ) \left (\int _1^x-\frac {\left (K[1]^4-2\right ) \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},i \sqrt {2} K[1]\right )}{\sqrt {2} \left (\operatorname {HermiteH}\left (-\frac {1}{2},i K[1]\right ) \operatorname {HermiteH}\left (\frac {1}{2},K[1]\right )+\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 )\right )}dK[1]+c_1\right )+\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},i \sqrt {2} x\right ) \left (\int _1^x\frac {\left (K[2]^4-2\right ) \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 ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**4 - x**2*y(x) + Derivative(y(x), (x, 2)) + 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)) + 2