60.1.517 problem 530
Internal
problem
ID
[10531]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
530
Date
solved
:
Wednesday, March 05, 2025 at 11:50:46 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.061 (sec). Leaf size: 414
ode:=diff(y(x),x)^3-y(x)*diff(y(x),x)^2+y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
x -6 \left (\int _{}^{y}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{{1}/{3}}}{4 \textit {\_a}^{2}+2 \textit {\_a} \left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{{1}/{3}}+\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{{2}/{3}}}d \textit {\_a} \right )-c_{1} &= 0 \\
\frac {12 \left (\int _{}^{y}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{{1}/{3}}}{\left (\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{{1}/{3}}-2 \textit {\_a} \right ) \left (i \sqrt {3}\, \textit {\_a} +\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{{1}/{3}}+\textit {\_a} \right )}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\
\frac {12 \left (\int _{}^{y}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{{1}/{3}}}{\left (-\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{{1}/{3}}+2 \textit {\_a} \right ) \left (-i \sqrt {3}\, \textit {\_a} +\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{{1}/{3}}+\textit {\_a} \right )}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+c_{1} -x}{i \sqrt {3}-1} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 68.751 (sec). Leaf size: 648
ode=y[x]^2 - y[x]*D[y[x],x]^2 + D[y[x],x]^3==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}}}{2 \sqrt [3]{2} K[1]^2+2 \sqrt [3]{2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}} K[1]+2^{2/3} \left (2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}\right )^{2/3}}dK[1]\&\right ]\left [\frac {x}{6}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}}}{2 i \sqrt [3]{2} \sqrt {3} K[2]^2-2 \sqrt [3]{2} K[2]^2+4 \sqrt [3]{2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}} K[2]-i 2^{2/3} \sqrt {3} \left (2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}\right )^{2/3}-2^{2/3} \left (2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}\right )^{2/3}}dK[2]\&\right ]\left [\frac {x}{12}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}}}{-2 i \sqrt [3]{2} \sqrt {3} K[3]^2-2 \sqrt [3]{2} K[3]^2+4 \sqrt [3]{2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}} K[3]+i 2^{2/3} \sqrt {3} \left (2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}\right )^{2/3}-2^{2/3} \left (2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}\right )^{2/3}}dK[3]\&\right ]\left [\frac {x}{12}+c_1\right ] \\
\end{align*}
✓ Sympy. Time used: 109.138 (sec). Leaf size: 561
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(y(x)**2 - y(x)*Derivative(y(x), x)**2 + Derivative(y(x), x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - 6 \left (\sqrt {3} - i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}}}{- 4 \sqrt [3]{2} y^{2} i + 2 \sqrt {3} y \sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}} - 2 y i \sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}} + 2^{\frac {2}{3}} \sqrt {3} \left (- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}\right )^{\frac {2}{3}} + 2^{\frac {2}{3}} i \left (- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}\right )^{\frac {2}{3}}}\, dy = C_{1} - x, \ - 6 \left (\sqrt {3} + i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}}}{4 \sqrt [3]{2} y^{2} i + 2 \sqrt {3} y \sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}} + 2 y i \sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}} + 2^{\frac {2}{3}} \sqrt {3} \left (- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}\right )^{\frac {2}{3}} - 2^{\frac {2}{3}} i \left (- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}\right )^{\frac {2}{3}}}\, dy = C_{1} - x, \ 6 \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}}}{2 \sqrt [3]{2} y^{2} - 2 y \sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}} + 2^{\frac {2}{3}} \left (- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}\right )^{\frac {2}{3}}}\, dy = C_{1} - x\right ]
\]