29.35.9 problem 1041
Internal
problem
ID
[5607]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
35
Problem
number
:
1041
Date
solved
:
Tuesday, March 04, 2025 at 10:42:36 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}-{y^{\prime }}^{2}+y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.122 (sec). Leaf size: 459
ode:=diff(y(x),x)^3-diff(y(x),x)^2+y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= 0 \\
-3 \,3^{{5}/{6}} 2^{{2}/{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{{1}/{3}}}{3^{{5}/{6}} 2^{{2}/{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{{1}/{3}}+3^{{2}/{3}} 2^{{1}/{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{{2}/{3}}+6}d \textit {\_a} \right )+x -c_{1} &= 0 \\
\frac {36 \,3^{{5}/{6}} 2^{{2}/{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{{1}/{3}}}{\left (3^{{5}/{6}} 2^{{2}/{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{{1}/{3}}-6\right ) \left (3 i \sqrt {3}+3^{{5}/{6}} 2^{{2}/{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{{1}/{3}}+3\right )}d \textit {\_a} \right )+\left (1+i \sqrt {3}\right ) \left (-c_{1} +x \right )}{1+i \sqrt {3}} &= 0 \\
\frac {36 \,3^{{5}/{6}} 2^{{2}/{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{{1}/{3}}}{3 \left (-\frac {3^{{5}/{6}} 2^{{2}/{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{{1}/{3}}}{3}+i \sqrt {3}-1\right ) \left (3^{{5}/{6}} 2^{{2}/{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{{1}/{3}}-6\right )}d \textit {\_a} \right )+\left (i \sqrt {3}-1\right ) \left (-c_{1} +x \right )}{i \sqrt {3}-1} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 93.029 (sec). Leaf size: 578
ode=(D[y[x],x])^3 - (D[y[x],x])^2 +y[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2}}{2^{2/3} \left (-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2\right )^{2/3}+2 \sqrt [3]{-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2}+2 \sqrt [3]{2}}dK[1]\&\right ]\left [\frac {x}{6}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2}}{-i 2^{2/3} \sqrt {3} \left (-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2\right )^{2/3}-2^{2/3} \left (-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2\right )^{2/3}+4 \sqrt [3]{-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2}+2 i \sqrt [3]{2} \sqrt {3}-2 \sqrt [3]{2}}dK[2]\&\right ]\left [\frac {x}{12}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2}}{i 2^{2/3} \sqrt {3} \left (-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2\right )^{2/3}-2^{2/3} \left (-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2\right )^{2/3}+4 \sqrt [3]{-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2}-2 i \sqrt [3]{2} \sqrt {3}-2 \sqrt [3]{2}}dK[3]\&\right ]\left [\frac {x}{12}+c_1\right ] \\
\end{align*}
✓ Sympy. Time used: 56.240 (sec). Leaf size: 520
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(y(x)**2 + Derivative(y(x), x)**3 - Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - 6 \left (\sqrt {3} - i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2}}{2^{\frac {2}{3}} \sqrt {3} \left (27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2\right )^{\frac {2}{3}} + 2^{\frac {2}{3}} i \left (27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2\right )^{\frac {2}{3}} + 2 \sqrt {3} \sqrt [3]{27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2} - 2 i \sqrt [3]{27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2} - 4 \sqrt [3]{2} i}\, dy = C_{1} - x, \ - 6 \left (\sqrt {3} + i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2}}{2^{\frac {2}{3}} \sqrt {3} \left (27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2\right )^{\frac {2}{3}} - 2^{\frac {2}{3}} i \left (27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2\right )^{\frac {2}{3}} + 2 \sqrt {3} \sqrt [3]{27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2} + 2 i \sqrt [3]{27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2} + 4 \sqrt [3]{2} i}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2}}{2^{\frac {2}{3}} \left (27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2\right )^{\frac {2}{3}} - 2 \sqrt [3]{27 y^{2} + 3 \sqrt {3} \sqrt {y^{2} \left (27 y^{2} - 4\right )} - 2} + 2 \sqrt [3]{2}}\, dy = C_{1} - \frac {x}{6}\right ]
\]