23.4.82 problem 82

Internal problem ID [6384]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 82
Date solved : Tuesday, September 30, 2025 at 02:54:55 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 61

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

\[ y = \frac {\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {-a}\, x}{2}\right ) \sqrt {2}\, c_2 a \sqrt {\pi }\, x -2 \,{\mathrm e}^{\frac {a \,x^{2}}{2}} c_2 \sqrt {-a}+2 c_1 x \sqrt {-a}}{2 x \sqrt {-a}} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 55

ode=(2 - a*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 \sqrt {\frac {\pi }{2}} \sqrt {a} c_1 \text {erfi}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )-\frac {c_1 e^{\frac {a x^2}{2}}}{x}+c_2 \end{align*}

✓ Sympy. Time used: 0.111 (sec). Leaf size: 42

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

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