23.2.156 problem 159
Internal
problem
ID
[5511]
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
:
159
Date
solved
:
Tuesday, September 30, 2025 at 12:46:04 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} x^{2} {y^{\prime }}^{2}-3 x y^{\prime } y+x^{3}+2 y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.182 (sec). Leaf size: 49
ode:=x^2*diff(y(x),x)^2-3*x*y(x)*diff(y(x),x)+x^3+2*y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -2 x^{{3}/{2}} \\
y &= 2 x^{{3}/{2}} \\
y &= \frac {x \left (c_1^{2}+4 x \right )}{2 c_1} \\
y &= \frac {x \left (c_1^{2} x +4\right )}{2 c_1} \\
\end{align*}
✓ Mathematica. Time used: 60.187 (sec). Leaf size: 961
ode=x^2 (D[y[x],x])^2-3 x y[x] D[y[x],x]+x^3+2 y[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {e^{-\frac {3 c_1}{2}} \left (2 \sqrt [3]{2} e^{3 c_1} x^3+\left (-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}\right )}{2^{2/3} \sqrt [3]{-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}}\\ y(x)&\to \frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}+i\right ) \left (-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}-i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}}\\ y(x)&\to -\frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}-i\right ) \left (-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}+i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}}\\ y(x)&\to \frac {e^{-\frac {3 c_1}{2}} \left (2 \sqrt [3]{2} e^{3 c_1} x^3+\left (4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}\right )}{2^{2/3} \sqrt [3]{4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}}\\ y(x)&\to \frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}+i\right ) \left (4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}-i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}}\\ y(x)&\to -\frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}-i\right ) \left (4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}+i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**3 + x**2*Derivative(y(x), x)**2 - 3*x*y(x)*Derivative(y(x), x) + 2*y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(-4*x**3 + y(x)**2) + 3*y(x))/(2*x) c