Internal problem ID [13497]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 13. Higher order equations: Extending first order concepts. Additional exercises
page 259
Problem number: 13.4 (f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {y^{2} y^{\prime \prime }+y^{\prime \prime }+2 y {y^{\prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 299
dsolve(y(x)^2*diff(y(x),x$2)+diff(y(x),x$2)+2*y(x)*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i \\ y \left (x \right ) &= i \\ y \left (x \right ) &= \frac {\left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {2}{3}}-4}{2 \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {2}{3}}+4 i \sqrt {3}-4}{4 \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {2}{3}} \sqrt {3}-\left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {2}{3}}+4 i \sqrt {3}+4}{4 \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 54.871 (sec). Leaf size: 307
DSolve[y[x]^2*y''[x]+y''[x]+2*y[x]*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2+\sqrt [3]{2} \left (3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}}{2 \sqrt [3]{2}}+\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}} \\ y(x)\to \frac {1-i \sqrt {3}}{2^{2/3} \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}}{2 \sqrt [3]{2}} \\ \end{align*}