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]
\[ \boxed {2 {y^{\prime }}^{5}+2 x y^{\prime }-y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 1311
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 {20 \sqrt {5}\, \sqrt {-\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{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 )}\, \left (-\frac {3 \left (i \sqrt {3}-1\right ) \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (\left (1+i \sqrt {3}\right ) x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}-90 c_{1} -6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right ) x \right )}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= -\frac {20 \sqrt {5}\, \sqrt {-\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{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 )}\, \left (-\frac {3 \left (i \sqrt {3}-1\right ) \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (\left (1+i \sqrt {3}\right ) x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}-90 c_{1} -6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right ) x \right )}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= -\frac {20 \sqrt {5}\, \left (-\frac {3 \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (1+i \sqrt {3}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (\left (i \sqrt {3}-1\right ) x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}+90 c_{1} +6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right ) x \right ) \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{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 )}}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= \frac {20 \sqrt {5}\, \left (-\frac {3 \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (1+i \sqrt {3}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (\left (i \sqrt {3}-1\right ) x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}+90 c_{1} +6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right ) x \right ) \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{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 )}}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= -\frac {\left (\frac {\left (3 c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{5}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+x \left (x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}+45 c_{1} +3 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )\right ) \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 )}}{25 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15 c_{1} +\sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= \frac {\left (\frac {\left (3 c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{5}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+x \left (x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}+45 c_{1} +3 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )\right ) \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 )}}{25 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15 c_{1} +\sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ \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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