75.20.28 problem 667

Internal problem ID [17091]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 667
Date solved : Monday, March 31, 2025 at 03:40:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x y^{\prime \prime }+2 y^{\prime }+y&=1 \end{align*}

With initial conditions

\begin{align*} y \left (\infty \right )&=1 \end{align*}

✓ Maple. Time used: 0.171 (sec). Leaf size: 5

ode:=4*x*diff(diff(y(x),x),x)+2*diff(y(x),x)+y(x) = 1; 
ic:=y(infinity) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = 1 \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 25

ode=4*x*D[y[x],{x,2}]+2*D[y[x],x]+y[x]==1; 
ic={y[Infinity]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \cos \left (\sqrt {x}\right )+c_2 \sin \left (\sqrt {x}\right )+1 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), (x, 2)) + y(x) + 2*Derivative(y(x), x) - 1,0) 
ics = {y(oo): 1} 
dsolve(ode,func=y(x),ics=ics)
 

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