Internal problem ID [11211]
Internal file name [OUTPUT/10197_Tuesday_December_06_2022_03_59_53_AM_86153348/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first.
Article 26. Equations solvable for \(x\). Page 55
Problem number: Ex 1.
ODE order: 1.
ODE degree: 3.
The type(s) of ODE detected by this program : "dAlembert"
Maple gives the following as the ode type
[[_homogeneous, `class A`], _dAlembert]
\[ \boxed {y^{\prime } y \left (2 {y^{\prime }}^{2}+3\right )=-x} \]
Let \(p=y^{\prime }\) the ode becomes \begin {align*} p y \left (2 p^{2}+3\right ) = -x \end {align*}
Solving for \(y\) from the above results in \begin {align*} y &= -\frac {x}{p \left (2 p^{2}+3\right )}\tag {1A} \end {align*}
This has the form \begin {align*} y=xf(p)+g(p)\tag {*} \end {align*}
Where \(f,g\) are functions of \(p=y'(x)\). The above ode is dAlembert ode which is now solved. Taking derivative of (*) w.r.t. \(x\) gives \begin {align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end {align*}
Comparing the form \(y=x f + g\) to (1A) shows that \begin {align*} f &= -\frac {1}{2 p^{3}+3 p}\\ g &= 0 \end {align*}
Hence (2) becomes \begin {align*} p +\frac {1}{2 p^{3}+3 p} = \frac {x \left (6 p^{2}+3\right ) p^{\prime }\left (x \right )}{\left (2 p^{3}+3 p \right )^{2}}\tag {2A} \end {align*}
The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives \begin {align*} p +\frac {1}{2 p^{3}+3 p} = 0 \end {align*}
Solving for \(p\) from the above gives \begin {align*} p&=\frac {i \sqrt {2}}{2}\\ p&=-\frac {i \sqrt {2}}{2}\\ p&=i\\ p&=-i \end {align*}
Substituting these in (1A) gives \begin {align*} y&=-i x\\ y&=i x\\ y&=-\frac {i \sqrt {2}\, x}{2}\\ y&=\frac {i \sqrt {2}\, x}{2} \end {align*}
The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in \begin {align*} p^{\prime }\left (x \right ) = \frac {\left (p \left (x \right )+\frac {1}{2 p \left (x \right )^{3}+3 p \left (x \right )}\right ) \left (2 p \left (x \right )^{3}+3 p \left (x \right )\right )^{2}}{x \left (6 p \left (x \right )^{2}+3\right )}\tag {3} \end {align*}
This ODE is now solved for \(p \left (x \right )\).
Inverting the above ode gives \begin {align*} \frac {d}{d p}x \left (p \right ) = \frac {x \left (p \right ) \left (6 p^{2}+3\right )}{\left (2 p^{3}+3 p \right )^{2} \left (p +\frac {1}{2 p^{3}+3 p}\right )}\tag {4} \end {align*}
This ODE is now solved for \(x \left (p \right )\).
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} \frac {d}{d p}x \left (p \right ) + p(p)x \left (p \right ) &= q(p) \end {align*}
Where here \begin {align*} p(p) &=-\frac {3}{p \left (2 p^{2}+3\right ) \left (p^{2}+1\right )}\\ q(p) &=0 \end {align*}
Hence the ode is \begin {align*} \frac {d}{d p}x \left (p \right )-\frac {3 x \left (p \right )}{p \left (p^{2}+1\right ) \left (2 p^{2}+3\right )} = 0 \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int -\frac {3}{p \left (2 p^{2}+3\right ) \left (p^{2}+1\right )}d p} \\ &= {\mathrm e}^{\frac {3 \ln \left (p^{2}+1\right )}{2}-\ln \left (p \right )-\ln \left (2 p^{2}+3\right )} \\ \end{align*} Which simplifies to \[ \mu = \frac {\left (p^{2}+1\right )^{\frac {3}{2}}}{2 p^{3}+3 p} \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \mu x &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \left (\frac {\left (p^{2}+1\right )^{\frac {3}{2}} x}{2 p^{3}+3 p}\right ) &= 0 \end {align*}
Integrating gives \begin {align*} \frac {\left (p^{2}+1\right )^{\frac {3}{2}} x}{2 p^{3}+3 p} &= c_{3} \end {align*}
Dividing both sides by the integrating factor \(\mu =\frac {\left (p^{2}+1\right )^{\frac {3}{2}}}{2 p^{3}+3 p}\) results in \begin {align*} x \left (p \right ) &= \frac {c_{3} \left (2 p^{3}+3 p \right )}{\left (p^{2}+1\right )^{\frac {3}{2}}} \end {align*}
which simplifies to \begin {align*} x \left (p \right ) &= \frac {c_{3} p \left (2 p^{2}+3\right )}{\left (p^{2}+1\right )^{\frac {3}{2}}} \end {align*}
Now we need to eliminate \(p\) between the above and (1A). One way to do this is by solving (1) for \(p\). This results in \begin {align*} p&=\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{2 y}-\frac {y}{{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}\\ p&=-\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{4 y}+\frac {y}{2 {\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{2 y}+\frac {y}{{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}\right )}{2}\\ p&=-\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{4 y}+\frac {y}{2 {\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{2 y}+\frac {y}{{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}\right )}{2} \end {align*}
Substituting the above in the solution for \(x\) found above gives \begin{align*} x&=-\frac {\left (-2 {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 \,2^{\frac {1}{3}} y^{2}\right ) c_{3} 2^{\frac {5}{6}} {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left (-2^{\frac {1}{3}} y^{2}+{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}\right )}{\sqrt {-\frac {2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 y^{2}}{{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}}}\, y^{3} \left (2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 y^{2}\right ) \left (x -\sqrt {x^{2}+2 y^{2}}\right )} \\ x&=\frac {2^{\frac {1}{3}} c_{3} \left (4 {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}+\left (1+i \sqrt {3}\right ) 2^{\frac {2}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 y^{2} \left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left (\left (\sqrt {3}+i\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}-2^{\frac {1}{3}} \left (i-\sqrt {3}\right ) y^{2}\right )}{\left (x -\sqrt {x^{2}+2 y^{2}}\right ) \sqrt {\frac {2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) \left (1+i \sqrt {3}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 y^{2} \left (i \sqrt {3}-1\right )}{{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}}}\, y^{3} \left (2^{\frac {1}{3}} \left (i-\sqrt {3}\right ) \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 \left (\sqrt {3}+i\right ) y^{2}\right )} \\ x&=\frac {\left (-4 {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}+\left (i \sqrt {3}-1\right ) 2^{\frac {2}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 \left (1+i \sqrt {3}\right ) y^{2} 2^{\frac {1}{3}}\right ) 2^{\frac {1}{3}} c_{3} {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left ({\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left (-i+\sqrt {3}\right )+y^{2} 2^{\frac {1}{3}} \left (\sqrt {3}+i\right )\right )}{\sqrt {\frac {\left (1-i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 \left (-i \sqrt {3}-1\right ) y^{2}}{{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}}}\, y^{3} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) \left (2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}} \left (\sqrt {3}+i\right )+2 y^{2} \left (-i+\sqrt {3}\right )\right )} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= -i x \\ \tag{2} y &= i x \\ \tag{3} y &= -\frac {i \sqrt {2}\, x}{2} \\ \tag{4} y &= \frac {i \sqrt {2}\, x}{2} \\ \tag{5} x &= -\frac {\left (-2 {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 \,2^{\frac {1}{3}} y^{2}\right ) c_{3} 2^{\frac {5}{6}} {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left (-2^{\frac {1}{3}} y^{2}+{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}\right )}{\sqrt {-\frac {2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 y^{2}}{{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}}}\, y^{3} \left (2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 y^{2}\right ) \left (x -\sqrt {x^{2}+2 y^{2}}\right )} \\ \tag{6} x &= \frac {2^{\frac {1}{3}} c_{3} \left (4 {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}+\left (1+i \sqrt {3}\right ) 2^{\frac {2}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 y^{2} \left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left (\left (\sqrt {3}+i\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}-2^{\frac {1}{3}} \left (i-\sqrt {3}\right ) y^{2}\right )}{\left (x -\sqrt {x^{2}+2 y^{2}}\right ) \sqrt {\frac {2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) \left (1+i \sqrt {3}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 y^{2} \left (i \sqrt {3}-1\right )}{{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}}}\, y^{3} \left (2^{\frac {1}{3}} \left (i-\sqrt {3}\right ) \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 \left (\sqrt {3}+i\right ) y^{2}\right )} \\ \tag{7} x &= \frac {\left (-4 {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}+\left (i \sqrt {3}-1\right ) 2^{\frac {2}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 \left (1+i \sqrt {3}\right ) y^{2} 2^{\frac {1}{3}}\right ) 2^{\frac {1}{3}} c_{3} {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left ({\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left (-i+\sqrt {3}\right )+y^{2} 2^{\frac {1}{3}} \left (\sqrt {3}+i\right )\right )}{\sqrt {\frac {\left (1-i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 \left (-i \sqrt {3}-1\right ) y^{2}}{{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}}}\, y^{3} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) \left (2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}} \left (\sqrt {3}+i\right )+2 y^{2} \left (-i+\sqrt {3}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = -i x \] Verified OK.
\[ y = i x \] Verified OK.
\[ y = -\frac {i \sqrt {2}\, x}{2} \] Verified OK.
\[ y = \frac {i \sqrt {2}\, x}{2} \] Verified OK.
\[ x = -\frac {\left (-2 {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 \,2^{\frac {1}{3}} y^{2}\right ) c_{3} 2^{\frac {5}{6}} {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left (-2^{\frac {1}{3}} y^{2}+{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}\right )}{\sqrt {-\frac {2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 y^{2}}{{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}}}\, y^{3} \left (2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 y^{2}\right ) \left (x -\sqrt {x^{2}+2 y^{2}}\right )} \] Warning, solution could not be verified
\[ x = \frac {2^{\frac {1}{3}} c_{3} \left (4 {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}+\left (1+i \sqrt {3}\right ) 2^{\frac {2}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 y^{2} \left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left (\left (\sqrt {3}+i\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}-2^{\frac {1}{3}} \left (i-\sqrt {3}\right ) y^{2}\right )}{\left (x -\sqrt {x^{2}+2 y^{2}}\right ) \sqrt {\frac {2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) \left (1+i \sqrt {3}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 y^{2} \left (i \sqrt {3}-1\right )}{{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}}}\, y^{3} \left (2^{\frac {1}{3}} \left (i-\sqrt {3}\right ) \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}-2 \left (\sqrt {3}+i\right ) y^{2}\right )} \] Warning, solution could not be verified
\[ x = \frac {\left (-4 {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}+\left (i \sqrt {3}-1\right ) 2^{\frac {2}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 \left (1+i \sqrt {3}\right ) y^{2} 2^{\frac {1}{3}}\right ) 2^{\frac {1}{3}} c_{3} {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left ({\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}} \left (-i+\sqrt {3}\right )+y^{2} 2^{\frac {1}{3}} \left (\sqrt {3}+i\right )\right )}{\sqrt {\frac {\left (1-i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}+2 \left (-i \sqrt {3}-1\right ) y^{2}}{{\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {2}{3}}}}\, y^{3} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) \left (2^{\frac {1}{3}} \left (x -\sqrt {x^{2}+2 y^{2}}\right ) {\left (\left (-x +\sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}} \left (\sqrt {3}+i\right )+2 y^{2} \left (-i+\sqrt {3}\right )\right )} \] Warning, solution could not be verified
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y \left (2 {y^{\prime }}^{2}+3\right )=-x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{2 y}-\frac {y}{{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}, y^{\prime }=-\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{4 y}+\frac {y}{2 {\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{2 y}+\frac {y}{{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}\right )}{2}, y^{\prime }=-\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{4 y}+\frac {y}{2 {\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{2 y}+\frac {y}{{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}\right )}{2}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{2 y}-\frac {y}{{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{4 y}+\frac {y}{2 {\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{2 y}+\frac {y}{{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}\right )}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{4 y}+\frac {y}{2 {\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}{2 y}+\frac {y}{{\left (\left (-2 x +2 \sqrt {x^{2}+2 y^{2}}\right ) y^{2}\right )}^{\frac {1}{3}}}\right )}{2} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: *** Sublevel 2 *** Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables trying simple symmetries for implicit equations <- symmetries for implicit equations successful`
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 776
dsolve(x+diff(y(x),x)*y(x)*(2*diff(y(x),x)^2+3)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {i \sqrt {2}\, x}{2} \\ y \left (x \right ) &= \frac {i \sqrt {2}\, x}{2} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {-2 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {2}{3}} \textit {\_a}^{2}+2 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {1}{3}} \textit {\_a}^{3}-{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {2}{3}}+\textit {\_a} {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {1}{3}}+\textit {\_a}^{2}}{{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {1}{3}} \left (2 \textit {\_a}^{4}+3 \textit {\_a}^{2}+1\right )}d \textit {\_a} +c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {2 i {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {2}{3}} \sqrt {3}\, \textit {\_a}^{2}+i {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {2}{3}} \sqrt {3}-2 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {2}{3}} \textit {\_a}^{2}-4 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {1}{3}} \textit {\_a}^{3}+i \sqrt {3}\, \textit {\_a}^{2}-{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {2}{3}}-2 \textit {\_a} {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {1}{3}}+\textit {\_a}^{2}}{{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {1}{3}} \left (2 \textit {\_a}^{4}+3 \textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 i {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {2}{3}} \sqrt {3}\, \textit {\_a}^{2}+i {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {2}{3}} \sqrt {3}+2 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {2}{3}} \textit {\_a}^{2}+4 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {1}{3}} \textit {\_a}^{3}+i \sqrt {3}\, \textit {\_a}^{2}+{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {2}{3}}+2 \textit {\_a} {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {1}{3}}-\textit {\_a}^{2}}{{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )}^{\frac {1}{3}} \left (2 \textit {\_a}^{4}+3 \textit {\_a}^{2}+1\right )}d \textit {\_a} \right )+2 c_{1} \right ) x \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x+y'[x]*y[x]*(2*(y'[x])^2+3)==0,y[x],x,IncludeSingularSolutions -> True]
Timed out