60.2.66 problem 642
Internal
problem
ID
[10640]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
642
Date
solved
:
Friday, March 14, 2025 at 02:17:23 AM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {\left (-y^{2}+4 a x \right )^{2}}{y} \end{align*}
✓ Maple. Time used: 0.033 (sec). Leaf size: 227
ode:=diff(y(x),x) = (-y(x)^2+4*a*x)^2/y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {2 \,{\mathrm e}^{4 a \,x^{2}+2 \sqrt {2}\, \sqrt {a}\, x} \sqrt {\left (c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1\right ) \sqrt {a}\, \left (c_{1} \left (\sqrt {a}\, x -\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+\sqrt {a}\, x +\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{-8 a \,x^{2}} {\mathrm e}^{-4 \sqrt {2}\, \sqrt {a}\, x}}}{c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1} \\
y &= -\frac {2 \,{\mathrm e}^{4 a \,x^{2}+2 \sqrt {2}\, \sqrt {a}\, x} \sqrt {\left (c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1\right ) \sqrt {a}\, \left (c_{1} \left (\sqrt {a}\, x -\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+\sqrt {a}\, x +\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{-8 a \,x^{2}} {\mathrm e}^{-4 \sqrt {2}\, \sqrt {a}\, x}}}{c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1} \\
\end{align*}
✓ Mathematica. Time used: 0.161 (sec). Leaf size: 147
ode=D[y[x],x] == (4*a*x - y[x]^2)^2/y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (-\frac {2 a K[2]}{K[2]^4-8 a x K[2]^2+16 a^2 x^2-2 a}-\int _1^x-\frac {4 a^2 \left (4 K[2]^3-16 a K[1] K[2]\right )}{\left (K[2]^4-8 a K[1] K[2]^2+16 a^2 K[1]^2-2 a\right )^2}dK[1]\right )dK[2]+\int _1^x\left (\frac {4 a^2}{y(x)^4-8 a K[1] y(x)^2+16 a^2 K[1]^2-2 a}+2 a\right )dK[1]=c_1,y(x)\right ]
\]
✓ Sympy. Time used: 12.268 (sec). Leaf size: 212
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-(4*a*x - y(x)**2)**2/y(x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt {\frac {a \left (- 4 x e^{2 \sqrt {2} \left (C_{1} + 2 a x\right ) \sqrt {\frac {1}{a}}} + 4 x + \sqrt {2} \sqrt {\frac {1}{a}} e^{2 \sqrt {2} \left (C_{1} + 2 a x\right ) \sqrt {\frac {1}{a}}} + \sqrt {2} \sqrt {\frac {1}{a}}\right )}{1 - e^{2 \sqrt {2} \left (C_{1} + 2 a x\right ) \sqrt {\frac {1}{a}}}}}, \ y{\left (x \right )} = - \sqrt {\frac {a \left (4 x e^{2 \sqrt {2} \left (C_{1} + 2 a x\right ) \sqrt {\frac {1}{a}}} - 4 x - \sqrt {2} \sqrt {\frac {1}{a}} e^{2 \sqrt {2} \left (C_{1} + 2 a x\right ) \sqrt {\frac {1}{a}}} - \sqrt {2} \sqrt {\frac {1}{a}}\right )}{e^{2 \sqrt {2} \left (C_{1} + 2 a x\right ) \sqrt {\frac {1}{a}}} - 1}}\right ]
\]