Internal problem ID [4265]
Internal file name [OUTPUT/3758_Sunday_June_05_2022_10_46_56_AM_16666367/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1043.
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}+b y = -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*} p +a = \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*} p +a = 0 \end {align*}
Solving for \(p\) from the above gives \begin {align*} p&=-a \end {align*}
Substituting these in (1A) gives \begin {align*} y&=\frac {a \left (2 a^{2}-b x \right )}{b} \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 )+a}{\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}}{p +a}\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 (p +a \right )}\,\mathop {\mathrm {d}p}}\\ &= \frac {5 p a -\frac {3 p^{2}}{2}-5 a^{2} \ln \left (p +a \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 {-30 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (\frac {\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+8 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a +4 a^{2}}{\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )+\left (30 \ln \left (2\right ) a^{2}+30 \ln \left (3\right ) a^{2}+7 a^{2}+6 c_{2} b \right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+\left (14 a^{3}+27 a b x -3 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}+27 b y\right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}+28 a^{4}-432 a^{2} b x +48 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a -432 y a b}{6 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \\ x&=\frac {112 i \sqrt {3}\, a^{4}-1728 i \sqrt {3}\, a^{2} b x +i \sqrt {3}\, \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}-64 i \sqrt {3}\, \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}+240 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (12\right )+576 i \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a -1728 i \sqrt {3}\, a b y-240 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (\frac {i \left (\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}-4 a^{2}\right ) \sqrt {3}-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+16 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}}{\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )+\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}+48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b c_{2} +56 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} a^{2}-64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}-112 a^{4}+1728 a^{2} b x -192 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a +1728 y a b}{48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \\ x&=\frac {-576 a \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}+i \left (-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}+64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}-112 a^{4}+1728 a^{2} b x +1728 y a b \right ) \sqrt {3}-112 a^{4}-64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}+8 \left (216 b x -\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \left (30 \ln \left (\frac {i \left (-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+4 a^{2}\right ) \sqrt {3}-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+16 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}}{12 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )-7\right )\right ) a^{2}+1728 y a b +\left (\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+48 c_{2} b \right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}}{48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {a \left (2 a^{2}-b x \right )}{b} \\ \tag{2} x &= \frac {-30 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (\frac {\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+8 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a +4 a^{2}}{\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )+\left (30 \ln \left (2\right ) a^{2}+30 \ln \left (3\right ) a^{2}+7 a^{2}+6 c_{2} b \right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+\left (14 a^{3}+27 a b x -3 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}+27 b y\right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}+28 a^{4}-432 a^{2} b x +48 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a -432 y a b}{6 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \\ \tag{3} x &= \frac {112 i \sqrt {3}\, a^{4}-1728 i \sqrt {3}\, a^{2} b x +i \sqrt {3}\, \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}-64 i \sqrt {3}\, \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}+240 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (12\right )+576 i \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a -1728 i \sqrt {3}\, a b y-240 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (\frac {i \left (\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}-4 a^{2}\right ) \sqrt {3}-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+16 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}}{\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )+\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}+48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b c_{2} +56 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} a^{2}-64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}-112 a^{4}+1728 a^{2} b x -192 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a +1728 y a b}{48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \\ \tag{4} x &= \frac {-576 a \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}+i \left (-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}+64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}-112 a^{4}+1728 a^{2} b x +1728 y a b \right ) \sqrt {3}-112 a^{4}-64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}+8 \left (216 b x -\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \left (30 \ln \left (\frac {i \left (-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+4 a^{2}\right ) \sqrt {3}-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+16 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}}{12 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )-7\right )\right ) a^{2}+1728 y a b +\left (\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+48 c_{2} b \right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}}{48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \\ \end{align*}
Verification of solutions
\[ y = \frac {a \left (2 a^{2}-b x \right )}{b} \] Verified OK.
\[ x = \frac {-30 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (\frac {\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+8 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a +4 a^{2}}{\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )+\left (30 \ln \left (2\right ) a^{2}+30 \ln \left (3\right ) a^{2}+7 a^{2}+6 c_{2} b \right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+\left (14 a^{3}+27 a b x -3 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}+27 b y\right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}+28 a^{4}-432 a^{2} b x +48 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a -432 y a b}{6 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be verified
\[ x = \frac {112 i \sqrt {3}\, a^{4}-1728 i \sqrt {3}\, a^{2} b x +i \sqrt {3}\, \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}-64 i \sqrt {3}\, \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}+240 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (12\right )+576 i \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a -1728 i \sqrt {3}\, a b y-240 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (\frac {i \left (\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}-4 a^{2}\right ) \sqrt {3}-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+16 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}}{\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )+\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}+48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b c_{2} +56 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} a^{2}-64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}-112 a^{4}+1728 a^{2} b x -192 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a +1728 y a b}{48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be verified
\[ x = \frac {-576 a \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}+i \left (-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}+64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}-112 a^{4}+1728 a^{2} b x +1728 y a b \right ) \sqrt {3}-112 a^{4}-64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}+8 \left (216 b x -\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \left (30 \ln \left (\frac {i \left (-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+4 a^{2}\right ) \sqrt {3}-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+16 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}}{12 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )-7\right )\right ) a^{2}+1728 y a b +\left (\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+48 c_{2} b \right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}}{48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-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: 132
dsolve(diff(y(x),x)^3-a*diff(y(x),x)^2+b*y(x)+a*b*x = 0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 a^{3}-5 \,{\mathrm e}^{\operatorname {RootOf}\left (-10 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+16 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -13 a^{2}-2 b x \right )} a^{2}+4 \,{\mathrm e}^{2 \operatorname {RootOf}\left (-10 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+16 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -13 a^{2}-2 b x \right )} a -{\mathrm e}^{3 \operatorname {RootOf}\left (-10 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+16 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -13 a^{2}-2 b x \right )}-a b x}{b} \]
✓ Solution by Mathematica
Time used: 0.63 (sec). Leaf size: 398
DSolve[(y'[x])^3 - a*(y'[x])^2 +b*y[x]+a*b*x==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=\frac {5 a \left (\frac {\sqrt [3]{2 a^3+\sqrt {\left (2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} a^2}{3 \sqrt [3]{2 a^3+\sqrt {\left (2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}+\frac {a}{3}\right )-\frac {3}{2} \left (\frac {\sqrt [3]{2 a^3+\sqrt {\left (2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} a^2}{3 \sqrt [3]{2 a^3+\sqrt {\left (2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}+\frac {a}{3}\right )^2-5 a^2 \log \left (\frac {\sqrt [3]{2 a^3+\sqrt {\left (2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} a^2}{3 \sqrt [3]{2 a^3+\sqrt {\left (2 a^3-27 a b x-27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}+\frac {4 a}{3}\right )}{b}+c_1\right \},y(x)\right ] \]