Internal problem ID [6793]
Internal file name [OUTPUT/6040_Tuesday_July_26_2022_05_05_00_AM_657202/index.tex]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 97. The p-discriminant equation.
EXERCISES Page 314
Problem number: 15.
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]
\[ \boxed {{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y=0} \]
Let \(p=y^{\prime }\) the ode becomes \begin {align*} p^{3}+x \,p^{2}-y = 0 \end {align*}
Solving for \(y\) from the above results in \begin {align*} y &= p^{3}+x \,p^{2}\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 &= p^{3} \end {align*}
Hence (2) becomes \begin {align*} -p^{2}+p = \left (3 p^{2}+2 x p \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*}
Substituting these in (1A) gives \begin {align*} y&=0\\ y&=1+x \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 )}{3 p \left (x \right )^{2}+2 p \left (x \right ) x}\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 {3 p^{2}+2 x \left (p \right ) p}{-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 {3 p}{p -1} \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 {3 p}{p -1} \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 {3 p}{p -1}\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 {3 p}{p -1}\right )\\ \mathrm {d} \left (\left (p -1\right )^{2} x\right ) &= \left (-3 p \left (p -1\right )\right )\, \mathrm {d} p \end {align*}
Integrating gives \begin {align*} \left (p -1\right )^{2} x &= \int {-3 p \left (p -1\right )\,\mathrm {d} p}\\ \left (p -1\right )^{2} x &= -p^{3}+\frac {3}{2} p^{2} + 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 {-p^{3}+\frac {3}{2} p^{2}}{\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 p^{3}+3 p^{2}+2 c_{1}}{2 \left (p -1\right )^{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 (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}-\frac {x}{3}\\ p&=-\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}\right )}{2}\\ p&=-\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}-\frac {x}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
Substituting the above in the solution for \(x\) found above gives \begin{align*} x&=\frac {24 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+96 \left (x +\frac {3}{2}\right ) \left (\left (\sqrt {3}\, \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 x^{3} y}-\frac {x^{6}}{2}+\frac {27 x^{3} y}{2}-\frac {243 y^{2}}{4}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (-\frac {5 \sqrt {3}\, \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right ) x \right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) {\left (\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-2 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+4 x^{2}-6 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )}^{2}} \\ x&=\frac {96 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+192 \left (x +\frac {3}{2}\right ) \left (\left (-3 \left (i+\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 x^{3} y}+\frac {\left (x^{6}-27 x^{3} y+\frac {243 y^{2}}{2}\right ) \left (1+i \sqrt {3}\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+x \left (-\frac {15 \left (i-\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right )\right )\right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+4 x^{2}+4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )^{2}} \\ x&=\frac {96 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-192 \left (\left (-3 \left (x^{3}-\frac {27 y}{4}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {27 y^{2}-4 x^{3} y}+\frac {\left (x^{6}-27 x^{3} y+\frac {243 y^{2}}{2}\right ) \left (i \sqrt {3}-1\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (-\frac {15 \left (i+\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right )\right ) x \right ) \left (x +\frac {3}{2}\right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-4 x^{2}-4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}-\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )^{2}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= 0 \\ \tag{2} y &= 1+x \\ \tag{3} x &= \frac {24 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+96 \left (x +\frac {3}{2}\right ) \left (\left (\sqrt {3}\, \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 x^{3} y}-\frac {x^{6}}{2}+\frac {27 x^{3} y}{2}-\frac {243 y^{2}}{4}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (-\frac {5 \sqrt {3}\, \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right ) x \right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) {\left (\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-2 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+4 x^{2}-6 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )}^{2}} \\ \tag{4} x &= \frac {96 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+192 \left (x +\frac {3}{2}\right ) \left (\left (-3 \left (i+\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 x^{3} y}+\frac {\left (x^{6}-27 x^{3} y+\frac {243 y^{2}}{2}\right ) \left (1+i \sqrt {3}\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+x \left (-\frac {15 \left (i-\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right )\right )\right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+4 x^{2}+4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )^{2}} \\ \tag{5} x &= \frac {96 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-192 \left (\left (-3 \left (x^{3}-\frac {27 y}{4}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {27 y^{2}-4 x^{3} y}+\frac {\left (x^{6}-27 x^{3} y+\frac {243 y^{2}}{2}\right ) \left (i \sqrt {3}-1\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (-\frac {15 \left (i+\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right )\right ) x \right ) \left (x +\frac {3}{2}\right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-4 x^{2}-4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}-\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )^{2}} \\ \end{align*}
Verification of solutions
\[ y = 0 \] Verified OK.
\[ y = 1+x \] Verified OK.
\[ x = \frac {24 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+96 \left (x +\frac {3}{2}\right ) \left (\left (\sqrt {3}\, \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 x^{3} y}-\frac {x^{6}}{2}+\frac {27 x^{3} y}{2}-\frac {243 y^{2}}{4}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (-\frac {5 \sqrt {3}\, \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right ) x \right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) {\left (\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-2 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+4 x^{2}-6 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )}^{2}} \] Warning, solution could not be verified
\[ x = \frac {96 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+192 \left (x +\frac {3}{2}\right ) \left (\left (-3 \left (i+\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {27 y}{4}\right ) \sqrt {27 y^{2}-4 x^{3} y}+\frac {\left (x^{6}-27 x^{3} y+\frac {243 y^{2}}{2}\right ) \left (1+i \sqrt {3}\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+x \left (-\frac {15 \left (i-\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+\left (i \sqrt {3}-1\right ) \left (x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right )\right )\right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+4 x^{2}+4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}+12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be verified
\[ x = \frac {96 \left (x^{3}+\frac {3 x^{2}}{2}-3 y+3 c_{1} \right ) \left (x^{3}-\frac {3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}}{2}-\frac {27 y}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-192 \left (\left (-3 \left (x^{3}-\frac {27 y}{4}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {27 y^{2}-4 x^{3} y}+\frac {\left (x^{6}-27 x^{3} y+\frac {243 y^{2}}{2}\right ) \left (i \sqrt {3}-1\right )}{2}\right ) \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}+\left (-\frac {15 \left (i+\frac {\sqrt {3}}{3}\right ) \left (x^{3}-\frac {54 y}{5}\right ) \sqrt {27 y^{2}-4 x^{3} y}}{2}+\left (1+i \sqrt {3}\right ) \left (x^{6}-\frac {81 x^{3} y}{2}+243 y^{2}\right )\right ) x \right ) \left (x +\frac {3}{2}\right )}{\left (2 x^{3}-3 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}-27 y\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-4 x^{2}-4 x \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}-\left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {2}{3}}-12 \left (108 y-8 x^{3}+12 \sqrt {3}\, \sqrt {27 y^{2}-4 x^{3} y}\right )^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be verified
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{3}+x {y^{\prime }}^{2}-y=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 (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}-\frac {x}{3}, y^{\prime }=-\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}-\frac {x}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}\right )}{2}, y^{\prime }=-\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}\right )}{2}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}-\frac {x}{3} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}-\frac {x}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}\right )}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{3 \left (108 y-8 x^{3}+12 \sqrt {81 y^{2}-12 x^{3} y}\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 dAlembert <- dAlembert successful`
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 994
dsolve(diff(y(x),x)^3+x*diff(y(x),x)^2-y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (4 x^{2}-2 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}+9\right )^{2} \left (4 x^{2}+4 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}+9\right )}{-1728 x^{3}-7776 x^{2}-11664 x +23328 c_{1} +1296 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}+5832} \\ y \left (x \right ) &= \frac {\left (\frac {\left (-i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}}{4}+\left (2 x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (x +\frac {3}{2}\right )^{2} \left (i \sqrt {3}-1\right )\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}} \left (i-\sqrt {3}\right )}{4}-i \left (-x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (x +\frac {3}{2}\right )^{2} \left (\sqrt {3}+i\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\ y \left (x \right ) &= \frac {\left (\frac {\left (i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}}}{4}-\left (-2 x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (-i \sqrt {3}-1\right ) \left (x +\frac {3}{2}\right )^{2}\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )}{4}+i \left (x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{\frac {1}{3}}+\left (x +\frac {3}{2}\right )^{2} \left (i-\sqrt {3}\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\ \end{align*}
✓ Solution by Mathematica
Time used: 84.497 (sec). Leaf size: 1516
DSolve[(y'[x])^3+x*(y'[x])^2-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-16 x^4+8 \left (\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}-12\right ) x^3-4 \left (\left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27\right ){}^{2/3}-9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}+54\right ) x^2+6 \left (72 c_1+2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27\right ){}^{2/3}+4 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}\right ) x+3 \left (4 c_1 \left (2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27\right ){}^{2/3}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}+54\right )+9 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27\right ){}^{2/3}+12 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+2 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )} \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}+27 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27}+81\right )}{24 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {-\left (\left (4 x^3+18 x^2+27 x-27 c_1\right ) (2 c_1+1)\right )}+27\right ){}^{2/3}} \\ y(x)\to \frac {1}{6} \left (-\frac {i \left (\sqrt {3}-i\right ) x (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}}+\frac {1}{16} \left (-\frac {i \left (\sqrt {3}-i\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}-4 x+6\right ){}^2+i \left (\sqrt {3}+i\right ) x \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}+2 (3-2 x) x-6 x+6 c_1\right ) \\ y(x)\to \frac {1}{6} \left (\frac {i \left (\sqrt {3}+i\right ) x (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}}+\frac {1}{16} \left (\frac {\left (1-i \sqrt {3}\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}}+\left (1+i \sqrt {3}\right ) \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}+4 x-6\right ){}^2-\left (1+i \sqrt {3}\right ) x \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {-\left ((1+2 c_1) \left (4 x^3+18 x^2+27 x-27 c_1\right )\right )}-54 x+27+108 c_1}+2 (3-2 x) x-6 x+6 c_1\right ) \\ \end{align*}