23.3.431 problem 436

Internal problem ID [6145]
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 : 436
Date solved : Tuesday, September 30, 2025 at 02:22:20 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 26

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

\[ y = c_2 \left (1+x \right )^{{3}/{2}} \sqrt {x}-\frac {16 \left (x^{2}+\frac {3}{2} x +\frac {3}{8}\right ) c_1}{3} \]

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to c_1 \sqrt {x} (x+1)^{3/2}-\frac {2}{3} c_2 \left (8 x^2+12 x+3\right ) \end{align*}

✓ Sympy. Time used: 0.234 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = \left (x + 1\right )^{2} \left (C_{1} \sqrt {\frac {x}{x + 1}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, 0 \\ \frac {3}{2} \end {matrix}\middle | {\frac {x}{x + 1}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} -2, - \frac {1}{2} \\ \frac {1}{2} \end {matrix}\middle | {\frac {x}{x + 1}} \right )}\right ) \]