23.3.459 problem 464

Internal problem ID [6173]
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 : 464
Date solved : Tuesday, September 30, 2025 at 02:24:05 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 39

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

\[ y = \frac {x \left (\frac {\operatorname {arcsinh}\left (x \right )}{4}+c_1 \right ) \sqrt {x^{2}+1}}{2}-\frac {3 x^{2}}{16}+\frac {c_1 \,\operatorname {arcsinh}\left (x \right )}{2}+\frac {\operatorname {arcsinh}\left (x \right )^{2}}{16}+c_2 -\frac {1}{16} \]

✓ Mathematica. Time used: 0.091 (sec). Leaf size: 77

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

\begin{align*} y(x)&\to \frac {1}{16} \left (8 c_1 \text {arcsinh}(x)+2 \sqrt {x^2+1} x \text {arctanh}\left (\frac {x}{\sqrt {x^2+1}}\right )+\text {arctanh}\left (\frac {x}{\sqrt {x^2+1}}\right )^2-3 x^2+8 c_1 \sqrt {x^2+1} x+16 c_2\right ) \end{align*}

✓ Sympy. Time used: 4.443 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{1} + C_{2} \left (x \sqrt {x^{2} + 1} + \operatorname {asinh}{\left (x \right )}\right ) - \frac {x^{2}}{8} + \frac {\int \sqrt {x^{2} + 1} \operatorname {asinh}{\left (x \right )}\, dx}{4} \]