23.4.51 problem 51

Internal problem ID [6353]
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 : 51
Date solved : Tuesday, September 30, 2025 at 02:52:23 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y^{\prime \prime }&=\left (a -x \right ) {y^{\prime }}^{3} \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 57

ode:=diff(diff(y(x),x),x) = (a-x)*diff(y(x),x)^3; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \ln \left (x -a +\sqrt {a^{2}-2 a x +x^{2}-c_1}\right )+c_2 \\ y &= -\ln \left (x -a +\sqrt {a^{2}-2 a x +x^{2}-c_1}\right )+c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.485 (sec). Leaf size: 61

ode=D[y[x],{x,2}] == (a - x)*D[y[x],x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2-\log \left (\sqrt {-2 a x+x^2-2 c_1}-a+x\right )\\ y(x)&\to \log \left (\sqrt {-2 a x+x^2-2 c_1}-a+x\right )+c_2 \end{align*}

✓ Sympy. Time used: 174.200 (sec). Leaf size: 122

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-(a - x)*Derivative(y(x), x)**3 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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