Internal problem ID [7089]
Internal file name [OUTPUT/6075_Sunday_June_05_2022_04_17_47_PM_71869709/index.tex]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 45.
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]
\[ \boxed {y^{\prime } y+x {y^{\prime }}^{3}=1} \]
Let \(p=y^{\prime }\) the ode becomes \begin {align*} x \,p^{3}+p y = 1 \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&=-1\\ p&=0 \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} + c_{1} \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 {c_{1}}{\left (p +1\right )^{2}} \end {align*}
which simplifies to \begin {align*} x \left (p \right ) &= \frac {2 c_{1} 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} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\frac {x \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 ) 2^{\frac {1}{3}} {\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 3^{\frac {1}{3}} \left (\frac {2 y^{2} c_{1}}{9}+x \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}} \left (-\frac {4 y c_{1}}{3}+x \right )\right ) 2^{\frac {2}{3}}}{\left (-2^{\frac {2}{3}} 3^{\frac {1}{3}} x y+{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2} \left (-2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} x y+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} 3^{\frac {1}{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 {\left (\left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) c_{1} \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 ) 2^{\frac {1}{3}} {\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 ) 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 (i 3^{\frac {1}{3}}-3^{\frac {5}{6}}\right ) y 2^{\frac {2}{3}}\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^{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}} \\ 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 {\left (\left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) c_{1} \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 ) 2^{\frac {1}{3}} {\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 (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) \left (\frac {2 y^{2} c_{1}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 2^{\frac {2}{3}}}{{\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}}+\left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} y\right )\right )}^{2} {\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}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= -x -1 \\ \tag{2} x &= \frac {54 x^{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}} \left (\frac {x \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 ) 2^{\frac {1}{3}} {\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 3^{\frac {1}{3}} \left (\frac {2 y^{2} c_{1}}{9}+x \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}} \left (-\frac {4 y c_{1}}{3}+x \right )\right ) 2^{\frac {2}{3}}}{\left (-2^{\frac {2}{3}} 3^{\frac {1}{3}} x y+{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2} \left (-2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} x y+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} 3^{\frac {1}{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 {\left (\left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) c_{1} \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 ) 2^{\frac {1}{3}} {\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 ) 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 (i 3^{\frac {1}{3}}-3^{\frac {5}{6}}\right ) y 2^{\frac {2}{3}}\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^{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}} \\ \tag{4} 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 {\left (\left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) c_{1} \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 ) 2^{\frac {1}{3}} {\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 (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) \left (\frac {2 y^{2} c_{1}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 2^{\frac {2}{3}}}{{\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}}+\left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} y\right )\right )}^{2} {\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}} \\ \end{align*}
Verification of solutions
\[ y = -x -1 \] Verified OK.
\[ x = \frac {54 x^{3} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\frac {x \left (\sqrt {\frac {4 y^{3}+27 x}{x}}\, c_{1} 3^{\frac {1}{6}}-2 \,3^{\frac {2}{3}} \left (y-\frac {3 c_{1}}{2}\right )\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{3}+\frac {2^{\frac {2}{3}} 3^{\frac {5}{6}} x^{2} \sqrt {\frac {4 y^{3}+27 x}{x}}}{3}+3 x 3^{\frac {1}{3}} \left (\frac {2 y^{2} c_{1}}{9}+x \right ) 2^{\frac {2}{3}}+{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (-\frac {4 y c_{1}}{3}+x \right )\right ) 2^{\frac {2}{3}}}{\left (-2^{\frac {2}{3}} 3^{\frac {1}{3}} x y+{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2} \left (-2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} x y+2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be verified
\[ x = -\frac {36 x^{3} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\left (-\frac {8 y c_{1}}{9}+\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {\left (\left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) c_{1} \sqrt {\frac {4 y^{3}+27 x}{x}}-6 \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) \left (y-\frac {3 c_{1}}{2}\right )\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {4 y^{3}+27 x}{x}}}{3}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (\frac {2 y^{2} c_{1}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) 2^{\frac {2}{3}}}{{\left (\left (-i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (i 3^{\frac {1}{3}}-3^{\frac {5}{6}}\right ) y 2^{\frac {2}{3}}\right )}^{2} {\left (\frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} y\right )\right )}^{2}} \] Warning, solution could not be verified
\[ x = \frac {36 x^{3} \left (\left (\frac {8 y c_{1}}{9}-\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {\left (\left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) c_{1} \sqrt {\frac {4 y^{3}+27 x}{x}}-6 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) \left (y-\frac {3 c_{1}}{2}\right )\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {4 y^{3}+27 x}{x}}}{3}+\left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) \left (\frac {2 y^{2} c_{1}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 2^{\frac {2}{3}}}{{\left (\frac {\left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} y\right )\right )}^{2} {\left (\left (-i+\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) y 2^{\frac {2}{3}}\right )}^{2}} \] Warning, solution could not be verified
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y+x {y^{\prime }}^{3}=1 \\ \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 (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}}}, y^{\prime }=-\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 {\mathrm {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}, y^{\prime }=-\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 {\mathrm {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}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\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}}} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\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 {\mathrm {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} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\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 {\mathrm {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} \\ \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: 0.047 (sec). Leaf size: 1817
dsolve(diff(y(x),x)*y(x)=1-x*(diff(y(x),x))^3,y(x), singsol=all)
\begin{align*} \frac {12 \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 )+\left (\frac {3^{\frac {2}{3}} 2^{\frac {1}{3}} \left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{6}+2^{\frac {2}{3}} 3^{\frac {1}{3}} y \left (x \right )^{2}\right ) x \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}} x^{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 (y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}}-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 (\sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}\, 2^{\frac {2}{3}} 3^{\frac {5}{6}} x -2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}} 3^{\frac {2}{3}} 2^{\frac {1}{3}} y \left (x \right )+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 {{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (-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 )+x \left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{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 y \left (x \right )^{2} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}\right )\right ) c_{1} x^{3}}{6 {\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}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{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}}+y \left (x \right ) \left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right ) x 2^{\frac {2}{3}}\right )}^{2}}+x +\frac {24 \,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}} x^{4} 3^{\frac {1}{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 (-i \sqrt {3}-1\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 ) 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}+\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}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2}} &= 0 \\ \frac {{\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} x^{3} \left (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 )+\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{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 y \left (x \right )^{2} \left (-3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}}\right ) x \right )}{6 {\left (\left (i \sqrt {3}-1\right ) {\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 {5}{6}}+3^{\frac {1}{3}}\right ) x 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 \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}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2}}+x +\frac {24 \,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}} x^{4} 3^{\frac {1}{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 (-3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}}}{2}\right )}{{\left (\left (\sqrt {3}+i\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 (3^{\frac {5}{6}}-i 3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} y \left (x \right ) x \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 \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}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 89.497 (sec). Leaf size: 20717
DSolve[y'[x]*y[x]==1-x*(y'[x])^3,y[x],x,IncludeSingularSolutions -> True]
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