8.25 problem 13.4 (f)

Internal problem ID [13496]
Internal file name [OUTPUT/12668_Friday_February_16_2024_12_04_50_AM_74199402/index.tex]

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.

The type(s) of ODE detected by this program : "second_order_integrable_as_is", "second_order_ode_missing_x", "exact nonlinear second order ode"

Maple gives the following as the ode type

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\[ \boxed {y^{2} y^{\prime \prime }+y^{\prime \prime }+2 y {y^{\prime }}^{2}=0} \]

8.25.1 Solving as second order integrable as is ode

Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (\left (1+y^{2}\right ) y^{\prime \prime }+2 y {y^{\prime }}^{2}\right )d x &= 0 \\ \left (1+y^{2}\right ) y^{\prime } = c_{1} \end {align*}

Which is now solved for \(y\). Integrating both sides gives \begin {align*} \int \frac {y^{2}+1}{c_{1}}d y &= x +c_{2}\\ \frac {y \left (y^{2}+3\right )}{3 c_{1}}&=x +c_{2} \end {align*}

Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\\ y_2&=-\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2}\\ y_3&=-\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}} \\ \tag{2} y &= -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ \tag{3} y &= -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}

Verification of solutions

\[ y = \frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}} \] Verified OK.

\[ y = -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.

\[ y = -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.

8.25.2 Solving as second order ode missing x ode

This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(y\) an independent variable. Using \begin {align*} y' &= p(y) \end {align*}

Then \begin {align*} y'' &= \frac {dp}{dx}\\ &= \frac {dy}{dx} \frac {dp}{dy}\\ &= p \frac {dp}{dy} \end {align*}

Hence the ode becomes \begin {align*} \left (y^{2}+1\right ) p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+2 y p \left (y \right )^{2} = 0 \end {align*}

Which is now solved as first order ode for \(p(y)\). In canonical form the ODE is \begin {align*} p' &= F(y,p)\\ &= f( y) g(p)\\ &= -\frac {2 y p}{y^{2}+1} \end {align*}

Where \(f(y)=-\frac {2 y}{y^{2}+1}\) and \(g(p)=p\). Integrating both sides gives \begin {align*} \frac {1}{p} \,dp &= -\frac {2 y}{y^{2}+1} \,d y\\ \int { \frac {1}{p} \,dp} &= \int {-\frac {2 y}{y^{2}+1} \,d y}\\ \ln \left (p \right )&=-\ln \left (y^{2}+1\right )+c_{1}\\ p&={\mathrm e}^{-\ln \left (y^{2}+1\right )+c_{1}}\\ &=\frac {c_{1}}{y^{2}+1} \end {align*}

For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} y^{\prime } = \frac {c_{1}}{1+y^{2}} \end {align*}

Integrating both sides gives \begin {align*} \int \frac {y^{2}+1}{c_{1}}d y &= x +c_{2}\\ \frac {y \left (y^{2}+3\right )}{3 c_{1}}&=x +c_{2} \end {align*}

Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\\ y_2&=-\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2}\\ y_3&=-\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}} \\ \tag{2} y &= -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ \tag{3} y &= -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}

Verification of solutions

\[ y = \frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}} \] Verified OK.

\[ y = -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.

\[ y = -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.

8.25.3 Solving as type second_order_integrable_as_is (not using ABC version)

Writing the ode as \[ \left (1+y^{2}\right ) y^{\prime \prime }+2 y {y^{\prime }}^{2} = 0 \] Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (\left (1+y^{2}\right ) y^{\prime \prime }+2 y {y^{\prime }}^{2}\right )d x &= 0 \\ y^{2} y^{\prime }+y^{\prime } = c_{1} \end {align*}

Which is now solved for \(y\). Integrating both sides gives \begin {align*} \int \frac {y^{2}+1}{c_{1}}d y &= x +c_{2}\\ \frac {y \left (y^{2}+3\right )}{3 c_{1}}&=x +c_{2} \end {align*}

Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\\ y_2&=-\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2}\\ y_3&=-\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}} \\ \tag{2} y &= -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ \tag{3} y &= -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}

Verification of solutions

\[ y = \frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}} \] Verified OK.

\[ y = -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.

\[ y = -\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 c_{1} c_{2} +12 c_{1} x +4 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.

8.25.4 Solving as exact nonlinear second order ode ode

An exact non-linear second order ode has the form \begin {align*} a_{2} \left (x , y, y^{\prime }\right ) y^{\prime \prime }+a_{1} \left (x , y, y^{\prime }\right ) y^{\prime }+a_{0} \left (x , y, y^{\prime }\right )&=0 \end {align*}

Where the following conditions are satisfied \begin {align*} \frac {\partial a_2}{\partial y} &= \frac {\partial a_1}{\partial y'}\\ \frac {\partial a_2}{\partial x} &= \frac {\partial a_0}{\partial y'}\\ \frac {\partial a_1}{\partial x} &= \frac {\partial a_0}{\partial y} \end {align*}

Looking at the the ode given we see that \begin {align*} a_2 &= 1+y^{2}\\ a_1 &= 2 y y^{\prime }\\ a_0 &= 0 \end {align*}

Applying the conditions to the above shows this is a nonlinear exact second order ode. Therefore it can be reduced to first order ode given by \begin {align*} \int {a_2\,d y'} + \int {a_1\,d y} + \int {a_0\,d x} &= c_{1}\\ \int {1+y^{2}\,d y'} + \int {2 y y^{\prime }\,d y} + \int {0\,d x} &= c_{1} \end {align*}

Which results in \begin {align*} \left (1+y^{2}\right ) y^{\prime }+y^{2} y^{\prime } = c_{1} \end {align*}

Which is now solved Integrating both sides gives \begin {align*} \int \frac {2 y^{2}+1}{c_{1}}d y &= x +c_{2}\\ \frac {\frac {2}{3} y^{3}+y}{c_{1}}&=x +c_{2} \end {align*}

Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}-\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}\\ y_2&=-\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{4}+\frac {1}{2 \left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}+\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}\right )}{2}\\ y_3&=-\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{4}+\frac {1}{2 \left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}+\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}-\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}} \\ \tag{2} y &= -\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{4}+\frac {1}{2 \left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}+\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}\right )}{2} \\ \tag{3} y &= -\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{4}+\frac {1}{2 \left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}+\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}

Verification of solutions

\[ y = \frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}-\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}} \] Verified OK.

\[ y = -\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{4}+\frac {1}{2 \left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}+\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.

\[ y = -\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{4}+\frac {1}{2 \left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}+\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.

8.25.5 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (1+y^{2}\right ) y^{\prime \prime }+2 y {y^{\prime }}^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Define new dependent variable}\hspace {3pt} u \\ {} & {} & u \left (x \right )=y^{\prime } \\ \bullet & {} & \textrm {Compute}\hspace {3pt} y^{\prime \prime } \\ {} & {} & u^{\prime }\left (x \right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Use chain rule on the lhs}\hspace {3pt} \\ {} & {} & y^{\prime } \left (\frac {d}{d y}u \left (y \right )\right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Substitute in the definition of}\hspace {3pt} u \\ {} & {} & u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )=y^{\prime \prime } \\ \bullet & {} & \textrm {Make substitutions}\hspace {3pt} y^{\prime }=u \left (y \right ),y^{\prime \prime }=u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & \left (y^{2}+1\right ) u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )+2 y u \left (y \right )^{2}=0 \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d y}u \left (y \right )=-\frac {2 y u \left (y \right )}{y^{2}+1} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d y}u \left (y \right )}{u \left (y \right )}=-\frac {2 y}{y^{2}+1} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} y \\ {} & {} & \int \frac {\frac {d}{d y}u \left (y \right )}{u \left (y \right )}d y =\int -\frac {2 y}{y^{2}+1}d y +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (u \left (y \right )\right )=-\ln \left (y^{2}+1\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (y \right ) \\ {} & {} & u \left (y \right )=\frac {{\mathrm e}^{c_{1}}}{y^{2}+1} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (y \right ) \\ {} & {} & u \left (y \right )=\frac {{\mathrm e}^{c_{1}}}{y^{2}+1} \\ \bullet & {} & \textrm {Revert to original variables with substitution}\hspace {3pt} u \left (y \right )=y^{\prime },y =y \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{c_{1}}}{1+y^{2}} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{c_{1}}}{1+y^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \left (1+y^{2}\right ) y^{\prime }={\mathrm e}^{c_{1}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (1+y^{2}\right ) y^{\prime }d x =\int {\mathrm e}^{c_{1}}d x +c_{2} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{3}}{3}+y={\mathrm e}^{c_{1}} x +c_{2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\left (12 \,{\mathrm e}^{c_{1}} x +12 c_{2} +4 \sqrt {4+9 x^{2} \left ({\mathrm e}^{c_{1}}\right )^{2}+18 x c_{2} {\mathrm e}^{c_{1}}+9 c_{2}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (12 \,{\mathrm e}^{c_{1}} x +12 c_{2} +4 \sqrt {4+9 x^{2} \left ({\mathrm e}^{c_{1}}\right )^{2}+18 x c_{2} {\mathrm e}^{c_{1}}+9 c_{2}^{2}}\right )^{\frac {1}{3}}} \end {array} \]

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
trying 2nd order JacobiSN 
<- 2nd_order Liouville successful`
 

✓ 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*}