29.35.8 problem 1040
Internal
problem
ID
[5606]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
35
Problem
number
:
1040
Date
solved
:
Monday, January 27, 2025 at 12:22:00 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}+{y^{\prime }}^{2}-y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.119 (sec). Leaf size: 425
dsolve(diff(y(x),x)^3+diff(y(x),x)^2-y(x) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 0 \\
-3 \sqrt {3}\, 2^{{1}/{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}}{3^{{2}/{3}} 2^{{2}/{3}}-\sqrt {3}\, 2^{{1}/{3}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}+3^{{1}/{3}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{2}/{3}}}d \textit {\_a} \right )+x -c_{1} &= 0 \\
\frac {12 \sqrt {3}\, 2^{{1}/{3}} \left (\int _{}^{y \left (x \right )}-\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (3^{{1}/{3}} 2^{{1}/{3}}+3^{{1}/{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (i 3^{{5}/{6}} 2^{{1}/{3}}+3^{{1}/{3}} 2^{{1}/{3}}-2 \,3^{{1}/{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}\right )}d \textit {\_a} \right )+\left (1+i \sqrt {3}\right ) \left (-c_{1} +x \right )}{1+i \sqrt {3}} &= 0 \\
\frac {12 i \sqrt {3}\, 2^{{1}/{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (3^{{1}/{3}} 2^{{1}/{3}}+3^{{1}/{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (i 3^{{5}/{6}} 2^{{1}/{3}}-3^{{1}/{3}} 2^{{1}/{3}}+2 \,3^{{1}/{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}\right )}d \textit {\_a} \right )+\left (-c_{1} +x \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 150.507 (sec). Leaf size: 510
DSolve[(D[y[x],x])^3 + (D[y[x],x])^2 -y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[1]+3 \sqrt {3} \sqrt {K[1] (27 K[1]-4)}+2}}{2^{2/3} \left (-27 K[1]+3 \sqrt {3} \sqrt {K[1] (27 K[1]-4)}+2\right )^{2/3}+2 \sqrt [3]{-27 K[1]+3 \sqrt {3} \sqrt {K[1] (27 K[1]-4)}+2}+2 \sqrt [3]{2}}dK[1]\&\right ]\left [-\frac {x}{6}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2}}{-i 2^{2/3} \sqrt {3} \left (-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2\right )^{2/3}+2^{2/3} \left (-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2\right )^{2/3}-4 \sqrt [3]{-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2}+2 i \sqrt [3]{2} \sqrt {3}+2 \sqrt [3]{2}}dK[2]\&\right ]\left [\frac {x}{12}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2}}{i 2^{2/3} \sqrt {3} \left (-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2\right )^{2/3}+2^{2/3} \left (-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2\right )^{2/3}-4 \sqrt [3]{-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2}-2 i \sqrt [3]{2} \sqrt {3}+2 \sqrt [3]{2}}dK[3]\&\right ]\left [\frac {x}{12}+c_1\right ] \\
\end{align*}