Internal problem ID [4362]
Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley.
2006
Section: Chapter 8, Ordinary differential equations. Section 13. Miscellaneous problems. page
466
Problem number: 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational, _Bernoulli]
Solve \begin {gather*} \boxed {3 y^{2} x^{3} y^{\prime }-y^{3} x^{2}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 105
dsolve(3*x^3*y(x)^2*diff(y(x),x)-x^2*y(x)^3=1,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\left (27 x^{3} c_{1}-9\right ) x \right )^{\frac {1}{3}}}{3 x} \\ y \relax (x ) = -\frac {\left (\left (27 x^{3} c_{1}-9\right ) x \right )^{\frac {1}{3}}}{6 x}-\frac {i \sqrt {3}\, \left (\left (27 x^{3} c_{1}-9\right ) x \right )^{\frac {1}{3}}}{6 x} \\ y \relax (x ) = -\frac {\left (\left (27 x^{3} c_{1}-9\right ) x \right )^{\frac {1}{3}}}{6 x}+\frac {i \sqrt {3}\, \left (\left (27 x^{3} c_{1}-9\right ) x \right )^{\frac {1}{3}}}{6 x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.759 (sec). Leaf size: 85
DSolve[3*x^3*y[x]^2*y'[x]-x^2*y[x]^3==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt [3]{-\frac {1}{3}} \sqrt [3]{-1+3 c_1 x^3}}{x^{2/3}} \\ y(x)\to \frac {\sqrt [3]{-\frac {1}{3}+c_1 x^3}}{x^{2/3}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{-\frac {1}{3}+c_1 x^3}}{x^{2/3}} \\ \end{align*}