Internal problem ID [8112]
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]
Solve \begin {gather*} \boxed {a \left (y^{\prime }\right )^{3}+b \left (y^{\prime }\right )^{2}+c y^{\prime }-y-d=0} \end {gather*}
✓ Solution by Maple
Time used: 0.291 (sec). Leaf size: 1285
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*} x -\left (\int _{}^{y \relax (x )}\frac {6 \,6^{\frac {1}{3}} a \left (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 c^{3} a -4 b^{3} d -b^{2} c^{2}}\, a +108 \textit {\_a} \,a^{2}+108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {1}{3}}}{6^{\frac {1}{3}} \left (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 c^{3} a -4 b^{3} d -b^{2} c^{2}}\, a +108 \textit {\_a} \,a^{2}+108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {2}{3}}-12 \,6^{\frac {1}{3}} a c +4 \,6^{\frac {1}{3}} b^{2}-4 b \left (\sqrt {3}\, \left (27 \sqrt {3}\, a^{2} \textit {\_a} +27 \sqrt {3}\, a^{2} d +9 \sqrt {3}\, a b c -2 \sqrt {3}\, b^{3}+9 \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 c^{3} a -4 b^{3} d -b^{2} c^{2}}\, a \right )\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}-\frac {12 \,6^{\frac {1}{3}} a \left (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 c^{3} a -4 b^{3} d -b^{2} c^{2}}\, a +108 \textit {\_a} \,a^{2}+108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {1}{3}}}{i \sqrt {3}\, 6^{\frac {1}{3}} \left (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 c^{3} a -4 b^{3} d -b^{2} c^{2}}\, a +108 \textit {\_a} \,a^{2}+108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {2}{3}}+12 i \sqrt {3}\, 6^{\frac {1}{3}} a c -4 i \sqrt {3}\, 6^{\frac {1}{3}} b^{2}+6^{\frac {1}{3}} \left (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 c^{3} a -4 b^{3} d -b^{2} c^{2}}\, a +108 \textit {\_a} \,a^{2}+108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {2}{3}}-12 \,6^{\frac {1}{3}} a c +4 \,6^{\frac {1}{3}} b^{2}+8 b \left (\sqrt {3}\, \left (27 \sqrt {3}\, a^{2} \textit {\_a} +27 \sqrt {3}\, a^{2} d +9 \sqrt {3}\, a b c -2 \sqrt {3}\, b^{3}+9 \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 c^{3} a -4 b^{3} d -b^{2} c^{2}}\, a \right )\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}\frac {12 \,6^{\frac {1}{3}} a \left (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 c^{3} a -4 b^{3} d -b^{2} c^{2}}\, a +108 \textit {\_a} \,a^{2}+108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {1}{3}}}{i \sqrt {3}\, 6^{\frac {1}{3}} \left (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 c^{3} a -4 b^{3} d -b^{2} c^{2}}\, a +108 \textit {\_a} \,a^{2}+108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {2}{3}}+12 i \sqrt {3}\, 6^{\frac {1}{3}} a c -4 i \sqrt {3}\, 6^{\frac {1}{3}} b^{2}-6^{\frac {1}{3}} \left (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 c^{3} a -4 b^{3} d -b^{2} c^{2}}\, a +108 \textit {\_a} \,a^{2}+108 a^{2} d +36 a b c -8 b^{3}\right )^{\frac {2}{3}}+12 \,6^{\frac {1}{3}} a c -4 \,6^{\frac {1}{3}} b^{2}-8 b \left (\sqrt {3}\, \left (27 \sqrt {3}\, a^{2} \textit {\_a} +27 \sqrt {3}\, a^{2} d +9 \sqrt {3}\, a b c -2 \sqrt {3}\, b^{3}+9 \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 c^{3} a -4 b^{3} d -b^{2} c^{2}}\, a \right )\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.419 (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*}