23.3.217 problem 219

Internal problem ID [5931]
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 : 219
Date solved : Tuesday, September 30, 2025 at 02:06:24 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

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

\[ y = c_1 +{\mathrm e}^{-\frac {x^{2}}{2}} c_2 \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 21

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

\begin{align*} y(x)&\to c_2-c_1 e^{-\frac {x^2}{2}} \end{align*}

✓ Sympy. Time used: 0.112 (sec). Leaf size: 49

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

\[ y{\left (x \right )} = C_{1} + x^{- \left (\operatorname {re}{\left (x\right )}\right )^{2} + \left (\operatorname {im}{\left (x\right )}\right )^{2} + 2} \left (C_{2} \sin {\left (2 \log {\left (x \right )} \left |{\operatorname {re}{\left (x\right )} \operatorname {im}{\left (x\right )}}\right | \right )} + C_{3} \cos {\left (2 \log {\left (x \right )} \operatorname {re}{\left (x\right )} \operatorname {im}{\left (x\right )} \right )}\right ) \]