Internal problem ID [6135]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number: 27.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{3}-2 y \left (y^{\prime }\right )^{2}+4 x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.277 (sec). Leaf size: 422
dsolve(x*diff(y(x),x)^3-2*y(x)*diff(y(x),x)^2+4*x^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {3 x^{\frac {4}{3}}}{2} \\ y \relax (x ) = \frac {3 \left (-\frac {x^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, x^{\frac {1}{3}}}{2}\right ) x}{2} \\ y \relax (x ) = \frac {3 \left (-\frac {x^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, x^{\frac {1}{3}}}{2}\right ) x}{2} \\ y \relax (x )-\RootOf \left (-4 \,\mathrm {csgn}\left (\frac {c_{1}^{2}+32 \textit {\_Z}}{c_{1}}\right ) \sqrt {-2 c_{1}^{3}+64 c_{1} \textit {\_Z}}\, c_{1}^{3}-\mathrm {csgn}\left (\frac {c_{1}^{2}+32 \textit {\_Z}}{c_{1}}\right ) \left (-2 c_{1}^{3}+64 c_{1} \textit {\_Z} \right )^{\frac {3}{2}}+64 c_{1}^{3} x +\left (-2 c_{1}^{3}+64 c_{1} \textit {\_Z} \right )^{\frac {3}{2}}\right ) = 0 \\ y \relax (x )-\RootOf \left (4 \,\mathrm {csgn}\left (\frac {c_{1}^{2}+32 \textit {\_Z}}{c_{1}}\right ) \sqrt {-2 c_{1}^{3}+64 c_{1} \textit {\_Z}}\, c_{1}^{3}+\mathrm {csgn}\left (\frac {c_{1}^{2}+32 \textit {\_Z}}{c_{1}}\right ) \left (-2 c_{1}^{3}+64 c_{1} \textit {\_Z} \right )^{\frac {3}{2}}+64 c_{1}^{3} x -\left (-2 c_{1}^{3}+64 c_{1} \textit {\_Z} \right )^{\frac {3}{2}}\right ) = 0 \\ y \relax (x )-\RootOf \left (c_{1} \left (\mathrm {csgn}\left (\frac {c_{1}^{2}+32 \textit {\_Z}}{c_{1}}\right ) \sqrt {2 c_{1}^{3}-64 c_{1} \textit {\_Z}}\, c_{1}^{2}+32 \,\mathrm {csgn}\left (\frac {c_{1}^{2}+32 \textit {\_Z}}{c_{1}}\right ) \sqrt {2 c_{1}^{3}-64 c_{1} \textit {\_Z}}\, \textit {\_Z} +\sqrt {2 c_{1}^{3}-64 c_{1} \textit {\_Z}}\, c_{1}^{2}-32 c_{1}^{2} x -32 \sqrt {2 c_{1}^{3}-64 c_{1} \textit {\_Z}}\, \textit {\_Z} \right )\right ) = 0 \\ y \relax (x )-\RootOf \left (c_{1} \left (\mathrm {csgn}\left (\frac {c_{1}^{2}+32 \textit {\_Z}}{c_{1}}\right ) \sqrt {2 c_{1}^{3}-64 c_{1} \textit {\_Z}}\, c_{1}^{2}+32 \,\mathrm {csgn}\left (\frac {c_{1}^{2}+32 \textit {\_Z}}{c_{1}}\right ) \sqrt {2 c_{1}^{3}-64 c_{1} \textit {\_Z}}\, \textit {\_Z} +\sqrt {2 c_{1}^{3}-64 c_{1} \textit {\_Z}}\, c_{1}^{2}+32 c_{1}^{2} x -32 \sqrt {2 c_{1}^{3}-64 c_{1} \textit {\_Z}}\, \textit {\_Z} \right )\right ) = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 81.531 (sec). Leaf size: 15120
DSolve[x*y'[x]^3-2*y[x]*y'[x]^2+4*x^2==0,y[x],x,IncludeSingularSolutions -> True]
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