29.34.24 problem 1026

Internal problem ID [5595]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 34
Problem number : 1026
Date solved : Monday, January 27, 2025 at 12:13:07 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} {y^{\prime }}^{3}-x y^{\prime }+a y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 813

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

\begin{align*} \frac {192 c_{1} 12^{-\frac {1}{a -1}} \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}} \left (a -\frac {3}{2}\right )^{2} {\left (\frac {i \sqrt {3}\, \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}}-12 i \sqrt {3}\, x -\left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}}-12 x}{\left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{1}/{3}}}\right )}^{\frac {1}{a -1}}+4 \left (a -\frac {1}{2}\right ) x \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}}+9 \left (1+i \sqrt {3}\right ) \left (a y \left (x \right )-\frac {\sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}}{9}\right ) \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{1}/{3}}+12 \left (i \sqrt {3}-1\right ) x^{2}}{\left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}} \left (4 a -6\right )} &= 0 \\ -\frac {9 \left (\frac {64 c_{1} \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}} \left (a -\frac {3}{2}\right )^{2} {\left (-\frac {i \sqrt {3}\, \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}}-12 i \sqrt {3}\, x +\left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}}+12 x}{12 \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{1}/{3}}}\right )}^{\frac {1}{a -1}}}{3}-\frac {4 \left (a -\frac {1}{2}\right ) x \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}}}{9}+\left (i \sqrt {3}-1\right ) \left (a y \left (x \right )-\frac {\sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}}{9}\right ) \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{1}/{3}}+\frac {4 x^{2} \left (1+i \sqrt {3}\right )}{3}\right )}{\left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}} \left (4 a -6\right )} &= 0 \\ -\frac {48 \left (c_{1} \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}} 6^{-\frac {1}{a -1}} \left (a -\frac {3}{2}\right )^{2} {\left (\frac {\left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}}+12 x}{\left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{1}/{3}}}\right )}^{\frac {1}{a -1}}-\frac {\left (a -\frac {1}{2}\right ) x \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}}}{24}+\left (\frac {3 a y \left (x \right )}{16}-\frac {\sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}}{48}\right ) \left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{1}/{3}}-\frac {x^{2}}{4}\right )}{\left (-108 a y \left (x \right )+12 \sqrt {81 a^{2} y \left (x \right )^{2}-12 x^{3}}\right )^{{2}/{3}} \left (2 a -3\right )} &= 0 \\ \end{align*}

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[(D[y[x],x])^3 -x D[y[x],x]+a y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

Timed out