Internal problem ID [8142]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 562.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [_dAlembert]
Solve \begin {gather*} \boxed {a \left (\left (y^{\prime }\right )^{3}+1\right )^{\frac {1}{3}}+b x y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.675 (sec). Leaf size: 3961
dsolve(a*(diff(y(x),x)^3+1)^(1/3)+b*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*} x -\left (\frac {2 b^{2} x^{2} y \relax (x ) \left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \relax (x )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \relax (x )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\right )^{\frac {1}{3}}-4 b x y \relax (x )^{2} a^{2}+a \left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \relax (x )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \relax (x )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\right )^{\frac {2}{3}}}{2 \left (b^{3} x^{3}+a^{3}\right ) \left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \relax (x )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \relax (x )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\right )^{\frac {1}{3}}}\right )^{-\frac {b}{b -1}} \left (\int _{}^{\frac {2 b^{2} x^{2} y \relax (x ) \left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \relax (x )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \relax (x )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\right )^{\frac {1}{3}}-4 b x y \relax (x )^{2} a^{2}+a \left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \relax (x )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \relax (x )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\right )^{\frac {2}{3}}}{2 \left (b^{3} x^{3}+a^{3}\right ) \left (-4 b^{6} x^{6}-8 a^{3} b^{3} x^{3}-4 b^{3} x^{3} y \relax (x )^{3}+4 \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\, b^{3} x^{3}-4 a^{6}+4 y \relax (x )^{3} a^{3}+4 a^{3} \sqrt {b^{6} x^{6}+2 a^{3} b^{3} x^{3}+2 b^{3} x^{3} y \relax (x )^{3}+a^{6}-2 y \relax (x )^{3} a^{3}+y \relax (x )^{6}}\right )^{\frac {1}{3}}}}-\frac {\textit {\_a}^{1+\frac {b}{b -1}} a}{\left (b -1\right ) \left (\textit {\_a}^{3}+1\right )^{\frac {2}{3}}}d \textit {\_a} +c_{1}\right ) = 0 \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.089 (sec). Leaf size: 84
DSolve[-y[x] + b*x*y'[x] + a*(1 + y'[x]^3)^(1/3)==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=K[1]^{\frac {b}{1-b}} \left (\frac {a \int \frac {K[1]^{\frac {2 b-1}{b-1}}}{\left (K[1]^3+1\right )^{2/3}}dK[1]}{1-b}+c_1\right ),y(x)=a \sqrt [3]{K[1]^3+1}+b x K[1]\right \},\{K[1],y(x)\}\right ] \]