Internal problem ID [8862]
Internal file name [OUTPUT/7797_Monday_June_06_2022_12_22_41_AM_57140036/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 528.
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 C`], _dAlembert]
\[ \boxed {{y^{\prime }}^{3}+a {y^{\prime }}^{2}+b y=-a b x} \]
Let \(p=y^{\prime }\) the ode becomes \begin {align*} a \,p^{2}+p^{3}+y b = -a b x \end {align*}
Solving for \(y\) from the above results in \begin {align*} y &= -a x -\frac {a \,p^{2}+p^{3}}{b}\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 &= -a\\ g &= -\frac {p^{2} \left (a +p \right )}{b} \end {align*}
Hence (2) becomes \begin {align*} a +p = \left (-\frac {2 p \left (a +p \right )}{b}-\frac {p^{2}}{b}\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*} a +p = 0 \end {align*}
Solving for \(p\) from the above gives \begin {align*} p&=-a \end {align*}
Substituting these in (1A) gives \begin {align*} y&=-a 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 {a +p \left (x \right )}{-\frac {2 p \left (x \right ) \left (a +p \left (x \right )\right )}{b}-\frac {p \left (x \right )^{2}}{b}}\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 {-\frac {2 p \left (a +p \right )}{b}-\frac {p^{2}}{b}}{a +p}\tag {4} \end {align*}
This ODE is now solved for \(x \left (p \right )\). Integrating both sides gives \begin {align*} x \left (p \right ) &= \int { -\frac {p \left (2 a +3 p \right )}{b \left (a +p \right )}\,\mathop {\mathrm {d}p}}\\ &= -\frac {-p a +\frac {3 p^{2}}{2}+a^{2} \ln \left (a +p \right )}{b}+c_{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 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{6}+\frac {2 a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}-\frac {a}{3}\\ p&=-\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{12}-\frac {a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}-\frac {a}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{6}-\frac {2 a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}\right )}{2}\\ p&=-\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{12}-\frac {a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}-\frac {a}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{6}-\frac {2 a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
Substituting the above in the solution for \(x\) found above gives \begin{align*} x&=\frac {\left (10 a^{3}+27 a b x +27 b y-3 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+24 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}\, a +6 a^{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} \ln \left (2\right )+6 a^{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} \ln \left (3\right )-20 a^{4}+\left (-216 x b -\left (6 \ln \left (\frac {{\left (\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+2 a \right )}^{2}}{\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )+5\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}\right ) a^{2}-216 y a b +6 c_{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b}{6 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \\ x&=\frac {8 \left (6 a^{2} \ln \left (12\right )+\left (-5-6 \ln \left (\left (-1+i \sqrt {3}\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+\frac {4 \left (-i a \sqrt {3}-a +2 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}\right ) a}{\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )\right ) a^{2}+6 c_{2} b \right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+32 \left (-1-i \sqrt {3}\right ) a^{3} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+288 a \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}+i \left (\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}-80 a^{4}-864 a^{2} b x -864 y a b \right ) \sqrt {3}+\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b}{48 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \\ x&=\frac {8 \left (\left (-5-6 \ln \left (\frac {-\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+8 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}-i \sqrt {3}\, \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+4 i \sqrt {3}\, a^{2}}{12 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )\right ) a^{2}+6 c_{2} b \right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+32 \left (-1+i \sqrt {3}\right ) a^{3} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}-288 a \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}+i \left (-\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b \right ) \sqrt {3}+\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b}{48 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= -a x \\ \tag{2} x &= \frac {\left (10 a^{3}+27 a b x +27 b y-3 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+24 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}\, a +6 a^{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} \ln \left (2\right )+6 a^{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} \ln \left (3\right )-20 a^{4}+\left (-216 x b -\left (6 \ln \left (\frac {{\left (\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+2 a \right )}^{2}}{\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )+5\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}\right ) a^{2}-216 y a b +6 c_{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b}{6 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \\ \tag{3} x &= \frac {8 \left (6 a^{2} \ln \left (12\right )+\left (-5-6 \ln \left (\left (-1+i \sqrt {3}\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+\frac {4 \left (-i a \sqrt {3}-a +2 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}\right ) a}{\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )\right ) a^{2}+6 c_{2} b \right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+32 \left (-1-i \sqrt {3}\right ) a^{3} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+288 a \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}+i \left (\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}-80 a^{4}-864 a^{2} b x -864 y a b \right ) \sqrt {3}+\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b}{48 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \\ \tag{4} x &= \frac {8 \left (\left (-5-6 \ln \left (\frac {-\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+8 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}-i \sqrt {3}\, \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+4 i \sqrt {3}\, a^{2}}{12 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )\right ) a^{2}+6 c_{2} b \right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+32 \left (-1+i \sqrt {3}\right ) a^{3} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}-288 a \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}+i \left (-\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b \right ) \sqrt {3}+\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b}{48 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \\ \end{align*}
Verification of solutions
\[ y = -a x \] Verified OK.
\[ x = \frac {\left (10 a^{3}+27 a b x +27 b y-3 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+24 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}\, a +6 a^{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} \ln \left (2\right )+6 a^{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} \ln \left (3\right )-20 a^{4}+\left (-216 x b -\left (6 \ln \left (\frac {{\left (\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+2 a \right )}^{2}}{\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )+5\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}\right ) a^{2}-216 y a b +6 c_{2} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b}{6 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \] Verified OK.
\[ x = \frac {8 \left (6 a^{2} \ln \left (12\right )+\left (-5-6 \ln \left (\left (-1+i \sqrt {3}\right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+\frac {4 \left (-i a \sqrt {3}-a +2 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}\right ) a}{\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )\right ) a^{2}+6 c_{2} b \right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+32 \left (-1-i \sqrt {3}\right ) a^{3} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}+288 a \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}+i \left (\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}-80 a^{4}-864 a^{2} b x -864 y a b \right ) \sqrt {3}+\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b}{48 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be verified
\[ x = \frac {8 \left (\left (-5-6 \ln \left (\frac {-\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+8 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}-i \sqrt {3}\, \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+4 i \sqrt {3}\, a^{2}}{12 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}}\right )\right ) a^{2}+6 c_{2} b \right ) \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}}+32 \left (-1+i \sqrt {3}\right ) a^{3} \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {1}{3}}-288 a \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}+i \left (-\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b \right ) \sqrt {3}+\left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {4}{3}}+80 a^{4}+864 a^{2} b x +864 y a b}{48 \left (-8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {b \left (a x +y\right ) \left (4 a^{3}+27 a b x +27 b y\right )}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be verified
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{3}+a {y^{\prime }}^{2}+b y=-a b 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 (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{6}+\frac {2 a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}-\frac {a}{3}, y^{\prime }=-\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{12}-\frac {a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}-\frac {a}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{6}-\frac {2 a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}\right )}{2}, y^{\prime }=-\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{12}-\frac {a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}-\frac {a}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{6}-\frac {2 a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}\right )}{2}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{6}+\frac {2 a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}-\frac {a}{3} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{12}-\frac {a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}-\frac {a}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{6}-\frac {2 a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}\right )}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{12}-\frac {a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}-\frac {a}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\right )^{\frac {1}{3}}}{6}-\frac {2 a^{2}}{3 \left (-108 a b x -108 b y-8 a^{3}+12 \sqrt {81 a^{2} b^{2} x^{2}+162 a \,b^{2} x y+12 a^{4} b x +81 y^{2} b^{2}+12 b y a^{3}}\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: 127
dsolve(diff(y(x),x)^3+a*diff(y(x),x)^2+b*y(x)+a*b*x=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-{\mathrm e}^{3 \operatorname {RootOf}\left (-2 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+8 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -5 a^{2}-2 b x \right )}+2 \,{\mathrm e}^{2 \operatorname {RootOf}\left (-2 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+8 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -5 a^{2}-2 b x \right )} a -{\mathrm e}^{\operatorname {RootOf}\left (-2 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+8 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -5 a^{2}-2 b x \right )} a^{2}-a x b}{b} \]
✓ Solution by Mathematica
Time used: 0.546 (sec). Leaf size: 398
DSolve[a*b*x + b*y[x] + a*y'[x]^2 + y'[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=-\frac {-a \left (\frac {\sqrt [3]{-2 a^3+\sqrt {\left (-2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} a^2}{3 \sqrt [3]{-2 a^3+\sqrt {\left (-2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}-\frac {a}{3}\right )+\frac {3}{2} \left (\frac {\sqrt [3]{-2 a^3+\sqrt {\left (-2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} a^2}{3 \sqrt [3]{-2 a^3+\sqrt {\left (-2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}-\frac {a}{3}\right )^2+a^2 \log \left (\frac {\sqrt [3]{-2 a^3+\sqrt {\left (-2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} a^2}{3 \sqrt [3]{-2 a^3+\sqrt {\left (-2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}+\frac {2 a}{3}\right )}{b}+c_1\right \},y(x)\right ] \]