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: 0.109 (sec). Leaf size: 800
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 x^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {3 x^{\frac {4}{3}} \left (i \sqrt {3}-1\right )}{4} \\ y \left (x \right ) &= \frac {c_{1}^{3}-128 x^{2}}{32 c_{1}} \\ y \left (x \right ) &= \frac {c_{1}^{3}+128 x^{2}}{32 c_{1}} \\ y \left (x \right ) &= \frac {c_{1} \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-x^{2} \left (c_{1}^{3}-864 x^{2}\right )}\right )^{\frac {1}{3}}}{96}+\frac {c_{1}^{3}}{96 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-x^{2} \left (c_{1}^{3}-864 x^{2}\right )}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96} \\ y \left (x \right ) &= \frac {c_{1} \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {x^{2} \left (c_{1}^{3}+864 x^{2}\right )}+1728 x^{2}\right )^{\frac {1}{3}}}{96}+\frac {c_{1}^{3}}{96 \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {x^{2} \left (c_{1}^{3}+864 x^{2}\right )}+1728 x^{2}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96} \\ y \left (x \right ) &= \frac {\left (c_{1} -\left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}\right ) c_{1} \left (i \left (\left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}+c_{1} \right ) \sqrt {3}-c_{1} +\left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}\right )}{192 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (c_{1}^{3}-1728 x^{2}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) c_{1}}{192}-\frac {\left (i \sqrt {3}\, c_{1} +c_{1} -2 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}\right ) c_{1}^{2}}{192 \left (c_{1}^{3}-1728 x^{2}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (c_{1} -\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}\right ) c_{1} \left (i \left (\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}+c_{1} \right ) \sqrt {3}-c_{1} +\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}\right )}{192 \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) c_{1}}{192}-\frac {\left (i \sqrt {3}\, c_{1} +c_{1} -2 \left (c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}\right ) c_{1}^{2}}{192 \left (c_{1}^{3}+24 \sqrt {2}\, \sqrt {3}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\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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