Internal problem ID [3600]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 30
Problem number: 871.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G], _rational, _dAlembert]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}-a y y^{\prime }+b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.104 (sec). Leaf size: 646
dsolve(x*diff(y(x),x)^2-a*y(x)*diff(y(x),x)+b = 0,y(x), singsol=all)
\begin{align*} \frac {c_{1} \left (2 \left (\frac {a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a^{3} y \relax (x )^{2}+2 \sqrt {a^{2} y \relax (x )^{2}-4 b x}\, \left (\frac {a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a^{2} y \relax (x )-\left (\frac {a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a^{2} y \relax (x )^{2}-\sqrt {a^{2} y \relax (x )^{2}-4 b x}\, \left (\frac {a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a y \relax (x )-4 \left (\frac {a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a b x +2 \left (\frac {a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} b x \right )}{\left (a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}\right )^{2}}+x -\frac {4 b \,x^{2}}{\left (2 a -1\right ) \left (a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}\right )^{2}} = 0 \\ \frac {c_{1} \left (-2 \left (-\frac {-a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a^{3} y \relax (x )^{2}+2 \sqrt {a^{2} y \relax (x )^{2}-4 b x}\, \left (-\frac {-a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a^{2} y \relax (x )+\left (-\frac {-a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a^{2} y \relax (x )^{2}-\sqrt {a^{2} y \relax (x )^{2}-4 b x}\, \left (-\frac {-a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a y \relax (x )+4 \left (-\frac {-a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a b x -2 \left (-\frac {-a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} b x \right )}{\left (-a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}\right )^{2}}+x -\frac {4 b \,x^{2}}{\left (2 a -1\right ) \left (-a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b x}\right )^{2}} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.847 (sec). Leaf size: 143
DSolve[x (y'[x])^2-a y[x] y'[x]+b==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {2 \left ((a-1) \log \left (\sqrt {a^2 y(x)^2-4 b x}+(a-1) y(x)\right )+a \log \left (\sqrt {a^2 y(x)^2-4 b x}-a y(x)\right )\right )}{2 a-1}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 \left ((a-1) \log \left (\sqrt {a^2 y(x)^2-4 b x}-a y(x)+y(x)\right )+a \log \left (\sqrt {a^2 y(x)^2-4 b x}+a y(x)\right )\right )}{2 a-1}=c_1,y(x)\right ] \\ \end{align*}