[next] [prev] [prev-tail] [tail] [up]

54.1.517 problem 530

Internal problem ID [11831]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 530
Date solved : Tuesday, September 30, 2025 at 11:12:56 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.041 (sec). Leaf size: 412
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 \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 (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{{2}/{3}}+2 \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} +4 \textit {\_a}^{2}}d \textit {\_a} -c_1 &= 0 \\ \frac {12 \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 (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 ) \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 )}d \textit {\_a} +i \left (x -c_1 \right ) \sqrt {3}+x -c_1}{1+i \sqrt {3}} &= 0 \\ \frac {-12 \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 (-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 ) \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 )}d \textit {\_a} +i \left (x -c_1 \right ) \sqrt {3}-x +c_1}{i \sqrt {3}-1} &= 0 \\ \end{align*}
✓ Mathematica. Time used: 65.656 (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: 71.179 (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)
\[ \left [ - 6 \left (\sqrt {3} - i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}}}{- 4 \sqrt [3]{2} y^{2} i + 2 \sqrt {3} y \sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}} - 2 y i \sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}} + 2^{\frac {2}{3}} \sqrt {3} \left (- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}\right )^{\frac {2}{3}} + 2^{\frac {2}{3}} i \left (- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}\right )^{\frac {2}{3}}}\, dy = C_{1} - x, \ - 6 \left (\sqrt {3} + i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}}}{4 \sqrt [3]{2} y^{2} i + 2 \sqrt {3} y \sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}} + 2 y i \sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}} + 2^{\frac {2}{3}} \sqrt {3} \left (- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}\right )^{\frac {2}{3}} - 2^{\frac {2}{3}} i \left (- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}\right )^{\frac {2}{3}}}\, dy = C_{1} - x, \ 6 \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}}}{2 \sqrt [3]{2} y^{2} - 2 y \sqrt [3]{- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}} + 2^{\frac {2}{3}} \left (- 2 y^{3} + 27 y^{2} + 3 \sqrt {3} \sqrt {y^{4} \left (27 - 4 y\right )}\right )^{\frac {2}{3}}}\, dy = C_{1} - x\right ] \]

[next] [prev] [prev-tail] [front] [up]