56.2.18 problem 17

Internal problem ID [8822]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 17
Date solved : Sunday, March 30, 2025 at 01:40:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 24

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

\[ y = {\mathrm e}^{x} \operatorname {AiryAi}\left (1+x \right ) c_2 +{\mathrm e}^{x} \operatorname {AiryBi}\left (1+x \right ) c_1 -x \]

✓ Mathematica. Time used: 5.484 (sec). Leaf size: 87

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

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

✗ Sympy

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

NotImplementedError : The given ODE x**2/2 + x*y(x)/2 + Derivative(y(x), x) - Derivative(y(x), (x, 2))/2 + 1 cannot be solved by the factorable group method