23.4.98 problem 98

Internal problem ID [6400]
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 : 98
Date solved : Tuesday, September 30, 2025 at 02:56:17 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 2+4 x y^{\prime }+x^{2} {y^{\prime }}^{2}+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 15

ode:=2+4*x*diff(y(x),x)+x^2*diff(y(x),x)^2+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \ln \left (\frac {-c_2 x +c_1}{x^{2}}\right ) \]

✓ Mathematica. Time used: 0.14 (sec). Leaf size: 17

ode=2 + 4*x*D[y[x],x] + x^2*D[y[x],x]^2 + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -2 \log (x)+\log (x+c_1)+c_2 \end{align*}

✓ Sympy. Time used: 1.423 (sec). Leaf size: 65

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

\[ \left [ y{\left (x \right )} = C_{1} - \frac {i \left (- 3 i \log {\left (x \right )} - i \log {\left (\tan ^{2}{\left (C_{2} + \frac {i \log {\left (x \right )}}{2} \right )} + 1 \right )}\right )}{2}, \ y{\left (x \right )} = C_{1} - \frac {i \left (- 3 i \log {\left (x \right )} - i \log {\left (\tan ^{2}{\left (C_{2} + \frac {i \log {\left (x \right )}}{2} \right )} + 1 \right )}\right )}{2}\right ] \]