29.5.24 problem 141
Internal
problem
ID
[4741]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
5
Problem
number
:
141
Date
solved
:
Sunday, March 30, 2025 at 03:52:19 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: 60.184 (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