23.3.342 problem 345

Internal problem ID [6056]
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 : 345
Date solved : Tuesday, September 30, 2025 at 02:20:41 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 25

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

\[ y = \frac {c_1 \left (x^{2}+1\right ) \arctan \left (x \right )}{2}+c_2 \,x^{2}+\frac {c_1 x}{2}+c_2 \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 30

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

\begin{align*} y(x)&\to \frac {1}{2} c_2 \left (\left (x^2+1\right ) \arctan (x)+x\right )+c_1 \left (x^2+1\right ) \end{align*}

✓ Sympy. Time used: 0.188 (sec). Leaf size: 65

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

\[ y{\left (x \right )} = \frac {\left (x^{2} + 1\right ) \left (C_{1} \sqrt {\frac {x^{2}}{x^{2} + 1}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {x^{2}}{x^{2} + 1}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} -1, 0 \\ \frac {1}{2} \end {matrix}\middle | {\frac {x^{2}}{x^{2} + 1}} \right )}\right ) \sqrt [4]{x^{2}}}{\sqrt {x}} \]