23.2.332 problem 374

Internal problem ID [5687]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 374
Date solved : Tuesday, September 30, 2025 at 01:57:46 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.075 (sec). Leaf size: 48

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

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

✓ Mathematica. Time used: 17.685 (sec). Leaf size: 10217

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

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✓ Sympy. Time used: 9.955 (sec). Leaf size: 153

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

\[ \left [ y{\left (x \right )} = C_{1} - x + \int \limits ^{- C_{2} + x} \frac {r^{2}}{- r^{2} + r \sqrt {r^{2} - 4 a^{2}} + 4 a^{2}}\, dr - \int \limits ^{- C_{2} + x} \frac {r \sqrt {- \left (- r + 2 a\right ) \left (r + 2 a\right )}}{- r^{2} + r \sqrt {r^{2} - 4 a^{2}} + 4 a^{2}}\, dr, \ y{\left (x \right )} = C_{1} - x + \int \limits ^{- C_{2} + x} \frac {r^{2}}{- r^{2} - r \sqrt {r^{2} - 4 a^{2}} + 4 a^{2}}\, dr + \int \limits ^{- C_{2} + x} \frac {r \sqrt {- \left (- r + 2 a\right ) \left (r + 2 a\right )}}{- r^{2} - r \sqrt {r^{2} - 4 a^{2}} + 4 a^{2}}\, dr\right ] \]