Internal problem ID [3599]
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]
\[ \boxed {x {y^{\prime }}^{2}-a y y^{\prime }+b=0} \]
✓ Solution by Maple
Time used: 0.063 (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 \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} a^{3} y \left (x \right )^{2}+2 {\left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} \sqrt {a^{2} y \left (x \right )^{2}-4 x b}\, a^{2} y \left (x \right )-{\left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} a^{2} y \left (x \right )^{2}-{\left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} \sqrt {a^{2} y \left (x \right )^{2}-4 x b}\, a y \left (x \right )-4 {\left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} a b x +2 {\left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} b x \right )}{\left (a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}\right )^{2}}+x -\frac {4 b \,x^{2}}{\left (2 a -1\right ) \left (a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}\right )^{2}} = 0 \\ \frac {c_{1} \left (-2 {\left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} a^{3} y \left (x \right )^{2}+2 {\left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} \sqrt {a^{2} y \left (x \right )^{2}-4 x b}\, a^{2} y \left (x \right )+{\left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} a^{2} y \left (x \right )^{2}-{\left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} \sqrt {a^{2} y \left (x \right )^{2}-4 x b}\, a y \left (x \right )+4 {\left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} a b x -2 {\left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}} b x \right )}{\left (-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}\right )^{2}}+x -\frac {4 b \,x^{2}}{\left (2 a -1\right ) \left (-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 x b}\right )^{2}} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.706 (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*}