1.528 problem 529

Internal problem ID [8109]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 529.
ODE order: 1.
ODE degree: 3.

CAS Maple gives this as type [_dAlembert]

Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{3}+x \left (y^{\prime }\right )^{2}-y=0} \end {gather*}

✓ Solution by Maple

Time used: 0.243 (sec). Leaf size: 1473

dsolve(diff(y(x),x)^3+x*diff(y(x),x)^2-y(x)=0,y(x), singsol=all)
 

\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (-\frac {1}{3} x -\frac {1}{9} x^{2}-\frac {1}{4}\right )}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}\right )^{3}+\left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (-\frac {1}{3} x -\frac {1}{9} x^{2}-\frac {1}{4}\right )}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}\right )^{2} x \\ y \relax (x ) = \left (-\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}\right )}{2}\right )^{3}+\left (-\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2} x \\ y \relax (x ) = \left (-\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}\right )}{2}\right )^{3}+\left (-\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 c_{1}-8 x^{3}+27+6 \sqrt {-48 c_{1} x^{3}-216 c_{1} x^{2}-24 x^{3}+324 c_{1}^{2}-324 c_{1} x -108 x^{2}+162 c_{1}-162 x}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2} x \\ \end{align*}

✓ Solution by Mathematica

Time used: 83.543 (sec). Leaf size: 1405

DSolve[-y[x] + x*y'[x]^2 + y'[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{24} \left (-4 x^2+2 x \left (6+\sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}\right )+3 \left (9+\sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}\right )+\frac {24 c_1 (2 x+3)^3-(2 x+3)^3 \left (-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )\right ){}^{2/3}+108 c_1 (2 x+3) \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}-6 \sqrt {6} (2 x+3) \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))} \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}+54 (2 x+3) \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}+216 c_1 \left (-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )\right ){}^{2/3}-12 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))} \left (-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )\right ){}^{2/3}+108 \left (-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )\right ){}^{2/3}}{(2 x+3)^3}\right ) \\ y(x)\to -\frac {2 x^2}{3}+\frac {1}{6} i \left (\sqrt {3}+i\right ) x \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}-\frac {i \left (\sqrt {3}-i\right ) (2 x+3)^2 x}{6 \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}}+\frac {1}{96} \left (-4 x-\frac {i \left (\sqrt {3}-i\right ) (2 x+3)^2}{\sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}+6\right ){}^2+c_1 \\ y(x)\to -\frac {2 x^2}{3}-\frac {1}{6} i \left (\sqrt {3}-i\right ) x \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}+\frac {i \left (\sqrt {3}+i\right ) (2 x+3)^2 x}{6 \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}}+\frac {1}{96} \left (-4 x+\frac {i \left (\sqrt {3}+i\right ) (2 x+3)^2}{\sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}}-i \left (\sqrt {3}-i\right ) \sqrt [3]{-2 x (2 x (2 x+9)+27)+3 \left (2 \sqrt {6} \sqrt {-((1+2 c_1) (x (2 x (2 x+9)+27)-27 c_1))}+9+36 c_1\right )}+6\right ){}^2+c_1 \\ y(x)\to 0 \\ \end{align*}