56.2.16 problem 16

Internal problem ID [8820]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 16
Date solved : Sunday, March 30, 2025 at 01:40:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-y^{\prime }-x y-x^{2}-1&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)-diff(y(x),x)-x*y(x)-x^2-1 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 4.22 (sec). Leaf size: 107

ode=D[y[x],{x,2}]-D[y[x],x]-x*y[x]-x^2-1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 - x*y(x) - Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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