Internal problem ID [4251]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1026.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {{y^{\prime }}^{3}-x y^{\prime }+y a=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 821
dsolve(diff(y(x),x)^3-x*diff(y(x),x)+a*y(x) = 0,y(x), singsol=all)
\begin{align*} -\frac {48 \left (6^{-\frac {1}{a -1}} c_{1} \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (a -\frac {3}{2}\right )^{2} {\left (\frac {\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}+12 x}{\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}}-\frac {x \left (a -\frac {1}{2}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}}{24}+\left (\frac {3 y \left (x \right ) a}{16}-\frac {\sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}}{48}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}-\frac {x^{2}}{4}\right )}{\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (2 a -3\right )} &= 0 \\ \frac {192 c_{1} \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (a -\frac {3}{2}\right )^{2} 12^{-\frac {1}{a -1}} {\left (\frac {i \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \sqrt {3}-12 i \sqrt {3}\, x -\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}-12 x}{\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}}+4 x \left (a -\frac {1}{2}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}+9 \left (1+i \sqrt {3}\right ) \left (y \left (x \right ) a -\frac {\sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}}{9}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}+12 \left (-1+i \sqrt {3}\right ) x^{2}}{\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (4 a -6\right )} &= 0 \\ -\frac {9 \left (\frac {64 c_{1} \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (a -\frac {3}{2}\right )^{2} {\left (-\frac {i \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \sqrt {3}-12 i \sqrt {3}\, x +\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}+12 x}{12 \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}}}{3}-\frac {4 x \left (a -\frac {1}{2}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}}{9}+\left (y \left (x \right ) a -\frac {\sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}}{9}\right ) \left (-1+i \sqrt {3}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}+\frac {4 \left (1+i \sqrt {3}\right ) x^{2}}{3}\right )}{\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (4 a -6\right )} &= 0 \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(y'[x])^3 -x y'[x]+a y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Timed out