54.1.363 problem 372
Internal
problem
ID
[11677]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
372
Date
solved
:
Tuesday, September 30, 2025 at 10:06:21 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{2}-4 y^{3}+a y+b&=0 \end{align*}
✓ Maple. Time used: 0.030 (sec). Leaf size: 229
ode:=diff(y(x),x)^2-4*y(x)^3+a*y(x)+b = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{2}/{3}}+3 a}{6 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}} \\
y &= \frac {-i \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{2}/{3}} \sqrt {3}+3 i \sqrt {3}\, a -\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{2}/{3}}-3 a}{12 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {-i \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{2}/{3}} \sqrt {3}+3 i \sqrt {3}\, a +\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{2}/{3}}+3 a}{12 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}} \\
y &= \operatorname {WeierstrassP}\left (x +c_1 , a , b\right ) \\
\end{align*}
✓ Mathematica. Time used: 0.307 (sec). Leaf size: 273
ode=b + a*y[x] - 4*y[x]^3 + D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \wp (x-c_1;a,b)\\ y(x)&\to \wp (x+c_1;a,b)\\ y(x)&\to \frac {\left (\sqrt {81 b^2-3 a^3}+9 b\right )^{2/3}+\sqrt [3]{3} a}{2\ 3^{2/3} \sqrt [3]{\sqrt {81 b^2-3 a^3}+9 b}}\\ y(x)&\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (\sqrt {81 b^2-3 a^3}+9 b\right )^{2/3}-\sqrt [6]{3} \left (\sqrt {3}+3 i\right ) a}{12 \sqrt [3]{\sqrt {81 b^2-3 a^3}+9 b}}\\ y(x)&\to \frac {\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (\sqrt {81 b^2-3 a^3}+9 b\right )^{2/3}-\sqrt [6]{3} \left (\sqrt {3}-3 i\right ) a}{12 \sqrt [3]{\sqrt {81 b^2-3 a^3}+9 b}} \end{align*}
✓ Sympy. Time used: 0.600 (sec). Leaf size: 41
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(a*y(x) + b - 4*y(x)**3 + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {4 y^{3} - y a - b}}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {4 y^{3} - y a - b}}\, dy = C_{1} + x\right ]
\]