Internal problem ID [3749]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1034.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{3}-2 y y^{\prime }+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.137 (sec). Leaf size: 245
dsolve(diff(y(x),x)^3-2*y(x)*diff(y(x),x)+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ x -\left (\int _{}^{y \relax (x )}\frac {6 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+24 \textit {\_a}}d \textit {\_a} \right )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}\frac {12 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{\left (1+i \sqrt {3}\right ) \left (12 i \textit {\_a} \sqrt {3}-\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+12 \textit {\_a} \right )}d \textit {\_a} \right )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}\frac {12 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{\left (-1+i \sqrt {3}\right ) \left (12 i \textit {\_a} \sqrt {3}+\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}-12 \textit {\_a} \right )}d \textit {\_a} \right )-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.359 (sec). Leaf size: 421
DSolve[(y'[x])^3 -2 y[x] y'[x]+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2}}{\sqrt [3]{2} \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}+4 \sqrt [3]{3} \text {$\#$1}}d\text {$\#$1}\&\right ]\left [\frac {x}{6^{2/3}}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2}}{\sqrt [3]{2} 3^{2/3} \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}-\sqrt [3]{2} \sqrt [6]{3} i \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}-12 \text {$\#$1}-4 i \text {$\#$1} \sqrt {3}}d\text {$\#$1}\&\right ]\left [x \text {Root}\left [248832 \text {$\#$1}^6+1\&,3\right ]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2}}{\sqrt [3]{2} 3^{2/3} \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}+\sqrt [3]{2} \sqrt [6]{3} i \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}-12 \text {$\#$1}+4 i \text {$\#$1} \sqrt {3}}d\text {$\#$1}\&\right ]\left [x \text {Root}\left [248832 \text {$\#$1}^6+1\&,4\right ]+c_1\right ] \\ y(x)\to 0 \\ \end{align*}