23.1.141 problem 144 (a)

Internal problem ID [4748]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 144 (a)
Date solved : Tuesday, September 30, 2025 at 08:30:17 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} 3 y^{\prime }&=x +\sqrt {x^{2}-3 y} \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 108

ode:=3*diff(y(x),x) = x+(x^2-3*y(x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {2 \left (x^{2}-3 y\right )^{{3}/{2}} \left (x^{2} c_1 y^{2}-4 c_1 y^{3}+1\right )+2 x \left (x^{2} c_1 y^{2}-4 c_1 y^{3}-1\right ) \left (x^{2}-\frac {9 y}{2}\right )}{\left (x^{2}-4 y\right ) y^{2} \left (x +\sqrt {x^{2}-3 y}\right )^{2} \left (-2 \sqrt {x^{2}-3 y}+x \right )} = 0 \]

✓ Mathematica. Time used: 60.114 (sec). Leaf size: 499

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

\begin{align*} y(x)&\to \frac {1}{12} \left (x^2+\frac {x \left (x^3+216 e^{3 c_1}\right )}{\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}}+\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}\right )\\ y(x)&\to \frac {1}{24} \left (2 x^2-\frac {i \left (\sqrt {3}-i\right ) x \left (x^3+216 e^{3 c_1}\right )}{\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}\right )\\ y(x)&\to \frac {1}{24} \left (2 x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3+216 e^{3 c_1}\right )}{\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -x/3 - sqrt(x**2 - 3*y(x))/3 + Derivative(y(x), x) cannot be solved by the factorable group method