75.12.5 problem 279
Internal
problem
ID
[16792]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
12.
Miscellaneous
problems.
Exercises
page
93
Problem
number
:
279
Date
solved
:
Thursday, March 13, 2025 at 08:48:00 AM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} 5 x y-4 y^{2}-6 x^{2}+\left (y^{2}-8 x y+\frac {5 x^{2}}{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.173 (sec). Leaf size: 439
ode:=5*x*y(x)-4*y(x)^2-6*x^2+(y(x)^2-8*x*y(x)+5/2*x^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\frac {\left (416 c_{1}^{3} x^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}}}{2}+\frac {27 x^{2} c_{1}^{2}}{\left (416 c_{1}^{3} x^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}}}+4 c_{1} x}{c_{1}} \\
y &= \frac {54 i \sqrt {3}\, c_{1}^{2} x^{2}-i \sqrt {3}\, \left (416 c_{1}^{3} x^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 c_{1}^{3} x^{3}+1}\right )^{{2}/{3}}-54 c_{1}^{2} x^{2}+16 c_{1} x \left (416 c_{1}^{3} x^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}}-\left (416 c_{1}^{3} x^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 c_{1}^{3} x^{3}+1}\right )^{{2}/{3}}}{4 \left (416 c_{1}^{3} x^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}} c_{1}} \\
y &= -\frac {54 i \sqrt {3}\, c_{1}^{2} x^{2}-i \sqrt {3}\, \left (416 c_{1}^{3} x^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 c_{1}^{3} x^{3}+1}\right )^{{2}/{3}}+54 c_{1}^{2} x^{2}-16 c_{1} x \left (416 c_{1}^{3} x^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}}+\left (416 c_{1}^{3} x^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 c_{1}^{3} x^{3}+1}\right )^{{2}/{3}}}{4 \left (416 c_{1}^{3} x^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 c_{1}^{3} x^{3}+1}\right )^{{1}/{3}} c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 37.682 (sec). Leaf size: 741
ode=(5*x*y[x]-4*y[x]^2-6*x^2)+(y[x]^2-8*x*y[x]+25/10*x^2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2^{2/3}}+\frac {27 x^2}{\sqrt [3]{2} \sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+4 x \\
y(x)\to -\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2\ 2^{2/3}}-\frac {27 \left (1+i \sqrt {3}\right ) x^2}{2 \sqrt [3]{2} \sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+4 x \\
y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2\ 2^{2/3}}-\frac {27 \left (1-i \sqrt {3}\right ) x^2}{2 \sqrt [3]{2} \sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+4 x \\
y(x)\to \frac {27\ 2^{2/3} x^2+8 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3} x+\sqrt [3]{2} \left (\sqrt {3898} \sqrt {x^6}+208 x^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3}} \\
y(x)\to \frac {27 i 2^{2/3} \sqrt {3} x^2-27\ 2^{2/3} x^2+16 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3} x-i \sqrt [3]{2} \sqrt {3} \left (\sqrt {3898} \sqrt {x^6}+208 x^3\right )^{2/3}-\sqrt [3]{2} \left (\sqrt {3898} \sqrt {x^6}+208 x^3\right )^{2/3}}{4 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3}} \\
y(x)\to \frac {\left (\sqrt {3898} \sqrt {x^6}+208 x^3\right )^{2/3} \text {Root}\left [\text {$\#$1}^3-16\&,3\right ]-54 \sqrt [3]{-1} 2^{2/3} x^2+16 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3} x}{4 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-6*x**2 + 5*x*y(x) + (5*x**2/2 - 8*x*y(x) + y(x)**2)*Derivative(y(x), x) - 4*y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out