23.3.523 problem 529

Internal problem ID [6237]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 529
Date solved : Tuesday, September 30, 2025 at 02:37:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 16

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

\[ y = \left (c_1 +c_2 \left (x +\ln \left (x \right )\right )\right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 21

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

\begin{align*} y(x)&\to \sqrt {x} (c_2 (x+\log (x))+c_1) \end{align*}

✓ Sympy. Time used: 0.330 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} + C_{2} \left (\log {\left (\frac {x}{x + 1} \right )} - \log {\left (\frac {x}{x + 1} - 1 \right )} - \frac {1}{\frac {x}{x + 1} - 1}\right )\right ) \left (x + 1\right ) {{}_{1}F_{0}\left (\begin {matrix} -1 \\ \end {matrix}\middle | {\frac {x}{x + 1}} \right )} \]