54.3.260 problem 1276

Internal problem ID [12555]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1276
Date solved : Wednesday, October 01, 2025 at 02:06:38 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (4 x^{2}+1\right ) y-4 \sqrt {x^{3}}\, {\mathrm e}^{x}&=0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 31

ode:=4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)-(4*x^2+1)*y(x)-4*(x^3)^(1/2)*exp(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\sinh \left (x \right ) c_2}{\sqrt {x}}+\frac {\cosh \left (x \right ) c_1}{\sqrt {x}}+\frac {\sqrt {x^{3}}\, {\mathrm e}^{x}}{2 x} \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 55

ode=-4*E^x*Sqrt[x^3] - (1 + 4*x^2)*y[x] + 4*x*D[y[x],x] + 4*x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^x \sqrt {x^3} (2 x-1)}{4 x^2}+\frac {c_1 e^{-x}}{\sqrt {x}}+\frac {c_2 e^x}{2 \sqrt {x}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), x) - (4*x**2 + 1)*y(x) - 4*sqrt(x**3)*exp(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*y(x) + x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) - sqrt(x**3)*exp(x)/x - y(x)/(4*x) cannot be solved by the factorable group method