Internal problem ID [2348]
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: 11.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {2 x {y^{\prime }}^{3}-{y^{\prime }}^{2} y=-1} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 513
dsolve(2*diff(y(x),x)^3*x+1=diff(y(x),x)^2*y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {9 \left (-\frac {2 \,3^{\frac {2}{3}} {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}} c_{1}^{2}}{9}+\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) \left (-\frac {2 \,3^{\frac {1}{3}} {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} c_{1}}{9}+x \right ) x \right ) 3^{\frac {1}{3}} x^{2}}{{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} \left (c_{1} 3^{\frac {1}{3}} x +{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}} \\ y \left (x \right ) &= \frac {4 \left (3 \left (-i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) c_{1}^{2} {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+\left (c_{1} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}+9 x \right ) x \left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right )\right ) 3^{\frac {1}{3}} x^{2}}{{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} {\left (\left (i-\sqrt {3}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+c_{1} x \left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right )\right )}^{2}} \\ y \left (x \right ) &= -\frac {4 \,3^{\frac {1}{3}} \left (-3 c_{1}^{2} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+\left (c_{1} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}-9 x \right ) x \left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right )\right ) x^{2}}{{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} {\left (\left (\sqrt {3}+i\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+c_{1} \left (-3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) x \right )}^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 151.15 (sec). Leaf size: 17695
DSolve[2*y'[x]^3*x+1==y'[x]^2*y[x],y[x],x,IncludeSingularSolutions -> True]
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