23.3.343 problem 346

Internal problem ID [6057]
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 : 346
Date solved : Tuesday, September 30, 2025 at 02:20:42 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 38

ode=a - x*D[y[x],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} \left (-a x^2+c_1 \text {arcsinh}(x)+c_1 \sqrt {x^2+1} x+2 c_2\right ) \end{align*}

✓ Sympy. Time used: 1.304 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{1} + C_{2} \left (x \sqrt {x^{2} + 1} + \operatorname {asinh}{\left (x \right )}\right ) - \frac {a x^{2}}{2} \]