Internal problem ID [8866]
Internal file name [OUTPUT/7801_Monday_June_06_2022_12_26_06_AM_44665759/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 532.
ODE order: 1.
ODE degree: 3.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {a {y^{\prime }}^{3}+b {y^{\prime }}^{2}+c y^{\prime }-y=d} \] Solving the given ode for \(y^{\prime }\) results in \(3\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\left (108 a^{2} y+12 \sqrt {3}\, \sqrt {27 y^{2} a^{2}+54 y a^{2} d +18 y a b c -4 y b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{6 a}-\frac {2 \left (3 c a -b^{2}\right )}{3 a \left (108 a^{2} y+12 \sqrt {3}\, \sqrt {27 y^{2} a^{2}+54 y a^{2} d +18 y a b c -4 y b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}-\frac {b}{3 a} \tag {1} \\ y^{\prime }&=-\frac {\left (108 a^{2} y+12 \sqrt {3}\, \sqrt {27 y^{2} a^{2}+54 y a^{2} d +18 y a b c -4 y b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{12 a}+\frac {3 c a -b^{2}}{3 a \left (108 a^{2} y+12 \sqrt {3}\, \sqrt {27 y^{2} a^{2}+54 y a^{2} d +18 y a b c -4 y b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}-\frac {b}{3 a}+\frac {i \sqrt {3}\, \left (\frac {\left (108 a^{2} y+12 \sqrt {3}\, \sqrt {27 y^{2} a^{2}+54 y a^{2} d +18 y a b c -4 y b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{6 a}+\frac {2 c a -\frac {2 b^{2}}{3}}{a \left (108 a^{2} y+12 \sqrt {3}\, \sqrt {27 y^{2} a^{2}+54 y a^{2} d +18 y a b c -4 y b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}\right )}{2} \tag {2} \\ y^{\prime }&=-\frac {\left (108 a^{2} y+12 \sqrt {3}\, \sqrt {27 y^{2} a^{2}+54 y a^{2} d +18 y a b c -4 y b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{12 a}+\frac {3 c a -b^{2}}{3 a \left (108 a^{2} y+12 \sqrt {3}\, \sqrt {27 y^{2} a^{2}+54 y a^{2} d +18 y a b c -4 y b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}-\frac {b}{3 a}-\frac {i \sqrt {3}\, \left (\frac {\left (108 a^{2} y+12 \sqrt {3}\, \sqrt {27 y^{2} a^{2}+54 y a^{2} d +18 y a b c -4 y b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{6 a}+\frac {2 c a -\frac {2 b^{2}}{3}}{a \left (108 a^{2} y+12 \sqrt {3}\, \sqrt {27 y^{2} a^{2}+54 y a^{2} d +18 y a b c -4 y b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}\right )}{2} \tag {3} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int _{}^{y}\frac {6 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}-2 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}-12 c a +4 b^{2}}d \textit {\_a} = x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {6 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}-2 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}-12 c a +4 b^{2}}d \textit {\_a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {6 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}-2 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}-12 c a +4 b^{2}}d \textit {\_a} = x +c_{1} \] Verified OK.
Solving equation (2)
Integrating both sides gives \begin {align*} \int _{}^{y}\frac {12 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}+12 i \sqrt {3}\, a c -4 i \sqrt {3}\, b^{2}-\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}-4 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}+12 c a -4 b^{2}}d \textit {\_a} = x +c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {12 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}+12 i \sqrt {3}\, a c -4 i \sqrt {3}\, b^{2}-\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}-4 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}+12 c a -4 b^{2}}d \textit {\_a} &= x +c_{2} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {12 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}+12 i \sqrt {3}\, a c -4 i \sqrt {3}\, b^{2}-\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}-4 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}+12 c a -4 b^{2}}d \textit {\_a} = x +c_{2} \] Verified OK.
Solving equation (3)
Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {12 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{12 i \sqrt {3}\, a c -4 i \sqrt {3}\, b^{2}+i \sqrt {3}\, \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}-12 c a +4 b^{2}+4 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}+\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}-\frac {12 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{12 i \sqrt {3}\, a c -4 i \sqrt {3}\, b^{2}+i \sqrt {3}\, \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}-12 c a +4 b^{2}+4 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}+\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}}d \textit {\_a} &= x +c_{3} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}-\frac {12 a \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}}{12 i \sqrt {3}\, a c -4 i \sqrt {3}\, b^{2}+i \sqrt {3}\, \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}-12 c a +4 b^{2}+4 b \left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {1}{3}}+\left (108 a^{2} \textit {\_a} +12 \sqrt {3}\, \sqrt {27 \textit {\_a}^{2} a^{2}+54 \textit {\_a} \,a^{2} d +18 \textit {\_a} a b c -4 \textit {\_a} \,b^{3}+27 a^{2} d^{2}+18 a b c d +4 a \,c^{3}-4 b^{3} d -b^{2} c^{2}}\, a +108 a^{2} d +36 c b a -8 b^{3}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Verified OK.
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 <- differential order: 1; missing x successful`
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 922
dsolve(a*diff(y(x),x)^3+b*diff(y(x),x)^2+c*diff(y(x),x)-y(x)-d=0,y(x), singsol=all)
\begin{align*} 3 \sqrt {3}\, 2^{\frac {1}{3}} a \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}}}{\sqrt {3}\, 2^{\frac {1}{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}} b -3^{\frac {1}{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {2}{3}}+3 \,3^{\frac {2}{3}} 2^{\frac {2}{3}} \left (a c -\frac {b^{2}}{3}\right )}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {12 \,2^{\frac {1}{3}} \sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}}}{-3 \,2^{\frac {1}{3}} \left (i-\frac {\sqrt {3}}{3}\right ) b \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}}+2 \,3^{\frac {1}{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {2}{3}}+9 \,2^{\frac {2}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) \left (a c -\frac {b^{2}}{3}\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (1+i \sqrt {3}\right )}{1+i \sqrt {3}} &= 0 \\ \frac {12 \,2^{\frac {1}{3}} \sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}}}{-3 \,2^{\frac {1}{3}} \left (i+\frac {\sqrt {3}}{3}\right ) b \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {1}{3}}-2 \,3^{\frac {1}{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a c b}{3}-\frac {2 b^{3}}{27}\right ) \sqrt {3}\right )^{\frac {2}{3}}+9 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {2}{3}} \left (a c -\frac {b^{2}}{3}\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (i \sqrt {3}-1\right )}{i \sqrt {3}-1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.362 (sec). Leaf size: 1064
DSolve[-d - y[x] + c*y'[x] + b*y'[x]^2 + a*y'[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}}}{2 \sqrt [3]{2} b^2+2 \sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}} b-6 \sqrt [3]{2} a c+2^{2/3} \left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}\right )^{2/3}}d\text {$\#$1}\&\right ]\left [-\frac {x}{6 a}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}}}{2 i \sqrt [3]{2} \sqrt {3} b^2+2 \sqrt [3]{2} b^2-4 \sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}} b-6 i \sqrt [3]{2} \sqrt {3} a c-6 \sqrt [3]{2} a c-i 2^{2/3} \sqrt {3} \left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}\right )^{2/3}+2^{2/3} \left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}\right )^{2/3}}d\text {$\#$1}\&\right ]\left [\frac {x}{12 a}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}}}{-2 i \sqrt [3]{2} \sqrt {3} b^2+2 \sqrt [3]{2} b^2-4 \sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}} b+6 i \sqrt [3]{2} \sqrt {3} a c-6 \sqrt [3]{2} a c+i 2^{2/3} \sqrt {3} \left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}\right )^{2/3}+2^{2/3} \left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}\right )^{2/3}}d\text {$\#$1}\&\right ]\left [\frac {x}{12 a}+c_1\right ] \\ y(x)\to -d \\ \end{align*}