56.2.28 problem 27

Internal problem ID [8832]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 27
Date solved : Sunday, March 30, 2025 at 01:41:07 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 63

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

\[ y = {\mathrm e}^{\frac {x}{2}} \left (-\pi \int {\mathrm e}^{-\frac {x}{2}} \operatorname {AiryBi}\left (\frac {1}{4}+x \right ) x^{3}d x \operatorname {AiryAi}\left (\frac {1}{4}+x \right )+\pi \int {\mathrm e}^{-\frac {x}{2}} \operatorname {AiryAi}\left (\frac {1}{4}+x \right ) x^{3}d x \operatorname {AiryBi}\left (\frac {1}{4}+x \right )+\operatorname {AiryAi}\left (\frac {1}{4}+x \right ) c_2 +\operatorname {AiryBi}\left (\frac {1}{4}+x \right ) c_1 \right ) \]

✓ Mathematica. Time used: 9.633 (sec). Leaf size: 103

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

\[ y(x)\to e^{x/2} \left (\operatorname {AiryAi}\left (x+\frac {1}{4}\right ) \int _1^x-e^{-\frac {K[1]}{2}} \pi \operatorname {AiryBi}\left (K[1]+\frac {1}{4}\right ) K[1]^3dK[1]+\operatorname {AiryBi}\left (x+\frac {1}{4}\right ) \int _1^xe^{-\frac {K[2]}{2}} \pi \operatorname {AiryAi}\left (K[2]+\frac {1}{4}\right ) K[2]^3dK[2]+c_1 \operatorname {AiryAi}\left (x+\frac {1}{4}\right )+c_2 \operatorname {AiryBi}\left (x+\frac {1}{4}\right )\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE x**3 + x*y(x) + Derivative(y(x), x) - Derivative(y(x), (x, 2)) cannot be solved by the factorable group method