Internal problem ID [2351]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 38, page 173
Problem number: 14.
ODE order: 1.
ODE degree: 5.
CAS Maple gives this as type [_dAlembert]
\[ \boxed {2 {y^{\prime }}^{5}+2 y^{\prime } x -y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 1505
dsolve(2*diff(y(x),x)^5+2*diff(y(x),x)*x=y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {{\left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x +\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}^{2} \sqrt {-5 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x +\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}}{2000 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )}-\frac {x \sqrt {-5 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x +\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}}{5 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}} y \left (x \right ) = \frac {{\left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x +\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}^{2} \sqrt {-5 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x +\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}}{600000 c_{1} +40000 \sqrt {20 x^{3}+225 c_{1}^{2}}}+\frac {x \sqrt {-5 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x +\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}}{5 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}} y \left (x \right ) = -\frac {{\left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x -\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 x \right )}^{2} \sqrt {5}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x -\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 x \right )}}{2000 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )}-\frac {x \sqrt {5}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x -\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 x \right )}}{5 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}} y \left (x \right ) = \frac {{\left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x -\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 x \right )}^{2} \sqrt {5}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x -\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 x \right )}}{600000 c_{1} +40000 \sqrt {20 x^{3}+225 c_{1}^{2}}}+\frac {x \sqrt {5}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x -\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 x \right )}}{5 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}} y \left (x \right ) = -\frac {{\left (\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}^{2} \sqrt {10}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}}{500 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )}-\frac {x \sqrt {10}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}}{5 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}} y \left (x \right ) = \frac {{\left (\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}^{2} \sqrt {10}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}}{150000 c_{1} +10000 \sqrt {20 x^{3}+225 c_{1}^{2}}}+\frac {x \sqrt {10}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}}{5 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}} \end{align*}
✓ Solution by Mathematica
Time used: 2.303 (sec). Leaf size: 2226
DSolve[2*y'[x]^5+2*y'[x]*x==y[x],y[x],x,IncludeSingularSolutions -> True]
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