50.2.41 problem 40

Internal problem ID [10131]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 40
Date solved : Saturday, October 04, 2025 at 04:33:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x^{3} y-x^{4}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x)-x^3*y(x)-x^4 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{5}, \frac {2 x^{{5}/{2}}}{5}\right ) c_2 +\sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{5}, \frac {2 x^{{5}/{2}}}{5}\right ) c_1 -x \]

✓ Mathematica. Time used: 0.116 (sec). Leaf size: 219

ode=D[y[x],{x,2}]-x^3*y[x]-x^4==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sqrt [5]{-1} \operatorname {Gamma}\left (\frac {6}{5}\right ) \left (-5^{2/5} \sqrt [5]{x^{5/2}} x^{15/2} \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {Hypergeometric0F1Regularized}\left (\frac {11}{5},\frac {x^5}{25}\right ) \operatorname {BesselI}\left (-\frac {1}{5},\frac {2 x^{5/2}}{5}\right )+5\ 5^{4/5} \left (x^{5/2}\right )^{4/5} \operatorname {BesselI}\left (\frac {1}{5},\frac {2 x^{5/2}}{5}\right )+5\ 5^{3/5} x^5 \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {BesselI}\left (-\frac {6}{5},\frac {2 x^{5/2}}{5}\right ) \operatorname {BesselI}\left (\frac {1}{5},\frac {2 x^{5/2}}{5}\right )\right )}{25 x^{3/2} \text {Root}\left [25 \text {$\#$1}^5+1\&,5\right ]}+\frac {c_1 \sqrt {x} \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {BesselI}\left (-\frac {1}{5},\frac {2 x^{5/2}}{5}\right )}{\sqrt [5]{5}}+\sqrt [5]{-\frac {1}{5}} c_2 \sqrt {x} \operatorname {Gamma}\left (\frac {6}{5}\right ) \operatorname {BesselI}\left (\frac {1}{5},\frac {2 x^{5/2}}{5}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**4 - x**3*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve -x**4 - x**3*y(x) + Derivative(y(x), (x, 2))