23.4.44 problem 44

Internal problem ID [6346]
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 : 44
Date solved : Tuesday, September 30, 2025 at 02:52:13 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 32

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

\[ y = \int \frac {{\mathrm e}^{-\int f \left (x \right )d x}}{\int {\mathrm e}^{-\int f \left (x \right )d x} g \left (x \right )d x +c_1}d x +c_2 \]

✓ Mathematica. Time used: 1.887 (sec). Leaf size: 66

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

\begin{align*} y(x)&\to \int _1^x\frac {\exp \left (\int _1^{K[3]}-f(K[1])dK[1]\right )}{c_1-\int _1^{K[3]}-\exp \left (\int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]}dK[3]+c_2 \end{align*}

✓ Sympy. Time used: 3.603 (sec). Leaf size: 54

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

\[ \left [ y{\left (x \right )} = C_{1} + \int \frac {e^{- \int f{\left (x \right )}\, dx}}{C_{2} + \int g{\left (x \right )} e^{- \int f{\left (x \right )}\, dx}\, dx}\, dx, \ y{\left (x \right )} = C_{1} + \int \frac {e^{- \int f{\left (x \right )}\, dx}}{C_{2} + \int g{\left (x \right )} e^{- \int f{\left (x \right )}\, dx}\, dx}\, dx\right ] \]