60.1.527 problem 530
Internal
problem
ID
[10541]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
530
Date
solved
:
Monday, January 27, 2025 at 08:52:35 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.061 (sec). Leaf size: 414
dsolve(diff(y(x),x)^3-y(x)*diff(y(x),x)^2+y(x)^2=0,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*}
✓ Solution by Mathematica
Time used: 68.751 (sec). Leaf size: 648
DSolve[y[x]^2 - y[x]*D[y[x],x]^2 + D[y[x],x]^3==0,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*}