Internal problem ID [6888]
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]]
\[ \boxed {x {y^{\prime }}^{3}-2 y {y^{\prime }}^{2}=-4 x^{2}} \]
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 831
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 \left (x \right ) = \frac {3 x^{\frac {4}{3}}}{2} y \left (x \right ) = \frac {3 \left (-\frac {x^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, x^{\frac {1}{3}}}{2}\right ) x}{2} y \left (x \right ) = \frac {3 \left (-\frac {x^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, x^{\frac {1}{3}}}{2}\right ) x}{2} y \left (x \right ) = -\frac {4 x^{2}}{c_{1}}+\frac {c_{1}^{2}}{32} y \left (x \right ) = \frac {4 x^{2}}{c_{1}}+\frac {c_{1}^{2}}{32} y \left (x \right ) = \frac {c_{1} \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {-6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}{96}+\frac {c_{1}^{3}}{96 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {-6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96} y \left (x \right ) = \frac {c_{1} \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}{96}+\frac {c_{1}^{3}}{96 \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96} y \left (x \right ) = -\frac {c_{1} \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {-6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}{192}-\frac {c_{1}^{3}}{192 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {-6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96}-\frac {i c_{1} \sqrt {3}\, \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {-6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}{192}+\frac {i \sqrt {3}\, c_{1}^{3}}{192 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {-6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}} y \left (x \right ) = -\frac {c_{1} \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {-6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}{192}-\frac {c_{1}^{3}}{192 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {-6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96}+\frac {i c_{1} \sqrt {3}\, \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {-6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}{192}-\frac {i \sqrt {3}\, c_{1}^{3}}{192 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {-6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}} y \left (x \right ) = -\frac {c_{1} \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}{192}-\frac {c_{1}^{3}}{192 \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96}-\frac {i c_{1} \sqrt {3}\, \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}{192}+\frac {i \sqrt {3}\, c_{1}^{3}}{192 \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}} y \left (x \right ) = -\frac {c_{1} \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}{192}-\frac {c_{1}^{3}}{192 \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96}+\frac {i c_{1} \sqrt {3}\, \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}}{192}-\frac {i \sqrt {3}\, c_{1}^{3}}{192 \left (1728 x^{2}+c_{1}^{3}+24 \sqrt {6 c_{1}^{3} x^{2}+5184 x^{4}}\right )^{\frac {1}{3}}} \end{align*}
✓ Solution by Mathematica
Time used: 171.698 (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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