Internal problem ID [15110]
Internal file name [OUTPUT/15111_Sunday_April_21_2024_01_29_58_PM_53052229/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 8.3. The Lagrange and Clairaut equations. Exercises page 72
Problem number: 223.
ODE order: 1.
ODE degree: 3.
The type(s) of ODE detected by this program : "dAlembert"
Maple gives the following as the ode type
[_dAlembert]
\[ \boxed {y-x {y^{\prime }}^{2}+\frac {1}{y^{\prime }}=0} \]
Let \(p=y^{\prime }\) the ode becomes \begin {align*} x \,p^{3}-y p -1 = 0 \end {align*}
Solving for \(y\) from the above results in \begin {align*} y &= p^{2} x -\frac {1}{p}\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 &= p^{2}\\ g &= -\frac {1}{p} \end {align*}
Hence (2) becomes \begin {align*} -p^{2}+p = \left (2 x p +\frac {1}{p^{2}}\right ) p^{\prime }\left (x \right )\tag {2A} \end {align*}
The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives \begin {align*} -p^{2}+p = 0 \end {align*}
Solving for \(p\) from the above gives \begin {align*} p&=0\\ p&=1 \end {align*}
Removing solutions for \(p\) which leads to undefined results and substituting these in (1A) gives \begin {align*} y&=x -1 \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 {-p \left (x \right )^{2}+p \left (x \right )}{2 p \left (x \right ) x +\frac {1}{p \left (x \right )^{2}}}\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 {2 x \left (p \right ) p +\frac {1}{p^{2}}}{-p^{2}+p}\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 {2}{p -1}\\ q(p) &=-\frac {1}{p^{3} \left (p -1\right )} \end {align*}
Hence the ode is \begin {align*} \frac {d}{d p}x \left (p \right )+\frac {2 x \left (p \right )}{p -1} = -\frac {1}{p^{3} \left (p -1\right )} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {2}{p -1}d p} \\ &= \left (p -1\right )^{2} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}}\left ( \mu x\right ) &= \left (\mu \right ) \left (-\frac {1}{p^{3} \left (p -1\right )}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \left (\left (p -1\right )^{2} x\right ) &= \left (\left (p -1\right )^{2}\right ) \left (-\frac {1}{p^{3} \left (p -1\right )}\right )\\ \mathrm {d} \left (\left (p -1\right )^{2} x\right ) &= \left (\frac {-p +1}{p^{3}}\right )\, \mathrm {d} p \end {align*}
Integrating gives \begin {align*} \left (p -1\right )^{2} x &= \int {\frac {-p +1}{p^{3}}\,\mathrm {d} p}\\ \left (p -1\right )^{2} x &= -\frac {1}{2 p^{2}}+\frac {1}{p} + \textit {\_C1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\left (p -1\right )^{2}\) results in \begin {align*} x \left (p \right ) &= \frac {-\frac {1}{2 p^{2}}+\frac {1}{p}}{\left (p -1\right )^{2}}+\frac {\textit {\_C1}}{\left (p -1\right )^{2}} \end {align*}
which simplifies to \begin {align*} x \left (p \right ) &= \frac {2 \textit {\_C1} \,p^{2}+2 p -1}{2 \left (p -1\right )^{2} p^{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 (12 \sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}+\frac {2 y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{\frac {1}{3}}}\\ p&=-\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{\frac {1}{3}}}{12 x}-\frac {y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}-\frac {2 y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}{2}\\ p&=-\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{\frac {1}{3}}}{12 x}-\frac {y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}-\frac {2 y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}{2} \end {align*}
Substituting the above in the solution for \(x\) found above gives \begin{align*} x&=-\frac {54 x^{3} \left (-\frac {x 2^{\frac {1}{3}} \left (\sqrt {\frac {-4 y^{3}+27 x}{x}}\, \textit {\_C1} 3^{\frac {1}{6}}+2 \,3^{\frac {2}{3}} \left (y+\frac {3 \textit {\_C1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{3}-\frac {2^{\frac {2}{3}} 3^{\frac {5}{6}} x^{2} \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}-3 x \left (\frac {2 y^{2} \textit {\_C1}}{9}+x \right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}}+\left (-\frac {4 y \textit {\_C1}}{3}+x \right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{\left (y 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2} \left (2 y 2^{\frac {1}{3}} 3^{\frac {2}{3}} x +2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}-6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2}} \\ x&=-\frac {36 x^{3} \left (\left (\frac {8 y \textit {\_C1}}{9}-\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {2^{\frac {1}{3}} \left (\textit {\_C1} \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) \left (y+\frac {3 \textit {\_C1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (\frac {2 y^{2} \textit {\_C1}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{{\left (-\frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} y\right )\right )}^{2} {\left (\left (\sqrt {3}+i\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (i 3^{\frac {1}{3}}-3^{\frac {5}{6}}\right ) y 2^{\frac {2}{3}}\right )}^{2}} \\ x&=\frac {36 x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\left (-\frac {8 y \textit {\_C1}}{9}+\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {2^{\frac {1}{3}} \left (\textit {\_C1} \left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) \left (y+\frac {3 \textit {\_C1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (\frac {2 y^{2} \textit {\_C1}}{9}+x \right ) \left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right )\right ) 2^{\frac {2}{3}}\right )\right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{{\left (\left (i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) y 2^{\frac {2}{3}}\right )}^{2} {\left (-\frac {\left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) y\right )\right )}^{2}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= x -1 \\ \tag{2} x &= -\frac {54 x^{3} \left (-\frac {x 2^{\frac {1}{3}} \left (\sqrt {\frac {-4 y^{3}+27 x}{x}}\, \textit {\_C1} 3^{\frac {1}{6}}+2 \,3^{\frac {2}{3}} \left (y+\frac {3 \textit {\_C1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{3}-\frac {2^{\frac {2}{3}} 3^{\frac {5}{6}} x^{2} \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}-3 x \left (\frac {2 y^{2} \textit {\_C1}}{9}+x \right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}}+\left (-\frac {4 y \textit {\_C1}}{3}+x \right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{\left (y 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2} \left (2 y 2^{\frac {1}{3}} 3^{\frac {2}{3}} x +2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}-6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2}} \\ \tag{3} x &= -\frac {36 x^{3} \left (\left (\frac {8 y \textit {\_C1}}{9}-\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {2^{\frac {1}{3}} \left (\textit {\_C1} \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) \left (y+\frac {3 \textit {\_C1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (\frac {2 y^{2} \textit {\_C1}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{{\left (-\frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} y\right )\right )}^{2} {\left (\left (\sqrt {3}+i\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (i 3^{\frac {1}{3}}-3^{\frac {5}{6}}\right ) y 2^{\frac {2}{3}}\right )}^{2}} \\ \tag{4} x &= \frac {36 x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\left (-\frac {8 y \textit {\_C1}}{9}+\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {2^{\frac {1}{3}} \left (\textit {\_C1} \left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) \left (y+\frac {3 \textit {\_C1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (\frac {2 y^{2} \textit {\_C1}}{9}+x \right ) \left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right )\right ) 2^{\frac {2}{3}}\right )\right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{{\left (\left (i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) y 2^{\frac {2}{3}}\right )}^{2} {\left (-\frac {\left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) y\right )\right )}^{2}} \\ \end{align*}
Verification of solutions
\[ y = x -1 \] Verified OK.
\[ x = -\frac {54 x^{3} \left (-\frac {x 2^{\frac {1}{3}} \left (\sqrt {\frac {-4 y^{3}+27 x}{x}}\, c_{1} 3^{\frac {1}{6}}+2 \,3^{\frac {2}{3}} \left (y+\frac {3 c_{1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{3}-\frac {2^{\frac {2}{3}} 3^{\frac {5}{6}} x^{2} \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}-3 x \left (\frac {2 y^{2} c_{1}}{9}+x \right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}}+\left (-\frac {4 y c_{1}}{3}+x \right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{\left (y 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2} \left (2 y 2^{\frac {1}{3}} 3^{\frac {2}{3}} x +2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}-6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be verified
\[ x = -\frac {36 x^{3} \left (\left (\frac {8 y c_{1}}{9}-\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {2^{\frac {1}{3}} \left (c_{1} \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) \left (y+\frac {3 c_{1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (\frac {2 y^{2} c_{1}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{{\left (-\frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} y\right )\right )}^{2} {\left (\left (\sqrt {3}+i\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (i 3^{\frac {1}{3}}-3^{\frac {5}{6}}\right ) y 2^{\frac {2}{3}}\right )}^{2}} \] Warning, solution could not be verified
\[ x = \frac {36 x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\left (-\frac {8 y c_{1}}{9}+\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {2^{\frac {1}{3}} \left (c_{1} \left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) \left (y+\frac {3 c_{1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (\frac {2 y^{2} c_{1}}{9}+x \right ) \left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right )\right ) 2^{\frac {2}{3}}\right )\right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{{\left (\left (i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) y 2^{\frac {2}{3}}\right )}^{2} {\left (-\frac {\left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) y\right )\right )}^{2}} \] Warning, solution could not be verified
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x {y^{\prime }}^{3}-y y^{\prime }-1=0 \\ \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 (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}+\frac {2 y}{{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}, y^{\prime }=-\frac {{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{12 x}-\frac {y}{{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}-\frac {2 y}{{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}{2}, y^{\prime }=-\frac {{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{12 x}-\frac {y}{{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}-\frac {2 y}{{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}{2}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}+\frac {2 y}{{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{12 x}-\frac {y}{{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}-\frac {2 y}{{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{12 x}-\frac {y}{{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{6 x}-\frac {2 y}{{\left (\left (108+12 \sqrt {-\frac {3 \left (4 y^{3}-27 x \right )}{x}}\right ) x^{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: -> 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 dAlembert <- dAlembert successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 1835
dsolve(y(x)=x*diff(y(x),x)^2-1/diff(y(x),x),y(x), singsol=all)
\begin{align*} \frac {12 x^{3} \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )+x \left (\frac {2^{\frac {1}{3}} \left (3^{\frac {1}{6}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+3 \,3^{\frac {2}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{2}+2^{\frac {2}{3}} 3^{\frac {1}{3}} y \left (x \right )^{2}\right )\right ) c_{1} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{\left (2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}+2 x \left (2^{\frac {1}{3}} 3^{\frac {2}{3}} y \left (x \right )-3 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )\right )^{2} \left (y \left (x \right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}}+x -\frac {18 x^{4} \left (2^{\frac {2}{3}} 3^{\frac {5}{6}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}\, x +2 y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+9 \,3^{\frac {1}{3}} 2^{\frac {2}{3}} x -3 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{\left (2 y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}} x +2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}-6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2} \left (y \left (x \right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}} &= 0 \\ -\frac {3 x^{3} \left (\frac {8 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )}{9}+x \left (\left (\left (\frac {i 3^{\frac {2}{3}}}{9}-\frac {3^{\frac {1}{6}}}{9}\right ) \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}-\frac {2 \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} y \left (x \right )^{2}}{9}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} c_{1}}{2 {\left (\left (i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}} y \left (x \right ) x \right )}^{2} {\left (-\frac {\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) y \left (x \right ) 2^{\frac {1}{3}}\right )\right )}^{2}}+x +\frac {216 x^{4} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (-{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y \left (x \right ) \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\frac {x \left (-i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) 2^{\frac {2}{3}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}}{2}+\frac {3 x \left (-i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{2}\right )}{{\left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (-i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) x y \left (x \right ) 2^{\frac {2}{3}}\right )}^{2} {\left (\frac {\left (-3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{2}+x \left (6 i {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {2}{3}}+3 \,3^{\frac {1}{6}}\right ) y \left (x \right ) 2^{\frac {1}{3}}\right )\right )}^{2}} &= 0 \\ \frac {3 x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} c_{1} \left (-\frac {8 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )}{9}+x \left (\left (\left (\frac {i 3^{\frac {2}{3}}}{9}+\frac {3^{\frac {1}{6}}}{9}\right ) \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}-\frac {2 \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} y \left (x \right )^{2}}{9}\right )\right )}{2 {\left (-\frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{6}+\left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} y \left (x \right ) \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2} {\left (\left (1-i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) y \left (x \right ) x \right )}^{2}}+x +\frac {216 x^{4} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}} \left ({\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y \left (x \right ) \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\frac {x \left (-i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) 2^{\frac {2}{3}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}}{2}+\frac {3 x \left (-i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{2}\right )}{\left (\frac {2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}} \left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right )}{2}+\left (6 i {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+y \left (x \right ) \left (i 3^{\frac {2}{3}}-3 \,3^{\frac {1}{6}}\right ) 2^{\frac {1}{3}}\right ) x \right )^{2} {\left ({\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\sqrt {3}+i\right )+\left (-3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) x y \left (x \right ) 2^{\frac {2}{3}}\right )}^{2}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 145.256 (sec). Leaf size: 19969
DSolve[y[x]==x*y'[x]^2-1/y'[x],y[x],x,IncludeSingularSolutions -> True]
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