Internal problem ID [8867]
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.
CAS Maple gives this as type [_quadrature]
\[ \boxed {a {y^{\prime }}^{3}+b {y^{\prime }}^{2}+y^{\prime } c -y=d} \]
✓ 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*}