Internal problem ID [4108]
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]
\[ \boxed {{y^{\prime }}^{2} x -a y y^{\prime }=-b} \]
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 381
dsolve(x*diff(y(x),x)^2-a*y(x)*diff(y(x),x)+b = 0,y(x), singsol=all)
\begin{align*} \frac {4 \left (2^{-\frac {1}{a -1}} y \left (x \right ) \left (a -\frac {1}{2}\right )^{2} a \sqrt {a^{2} y \left (x \right )^{2}-4 b x}-\frac {b x 2^{\frac {a -2}{a -1}}}{4}+2^{-\frac {1}{a -1}} \left (\left (a -\frac {1}{2}\right )^{2} a y \left (x \right )^{2}-2 b x \left (a -1\right )\right ) a \right ) c_{1} {\left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{x}\right )}^{\frac {1}{a -1}}+4 x \left (y \left (x \right ) \left (a -\frac {1}{2}\right ) \sqrt {a^{2} y \left (x \right )^{2}-4 b x}+\left (a^{2}-\frac {1}{2} a \right ) y \left (x \right )^{2}-2 b x \right ) a}{\left (2 a -1\right ) \left (a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}\right )^{2}} &= 0 \\ \frac {-4 c_{1} \left (-2^{-\frac {1}{a -1}} y \left (x \right ) \left (a -\frac {1}{2}\right )^{2} a \sqrt {a^{2} y \left (x \right )^{2}-4 b x}-\frac {b x 2^{\frac {a -2}{a -1}}}{4}+2^{-\frac {1}{a -1}} \left (\left (a -\frac {1}{2}\right )^{2} a y \left (x \right )^{2}-2 b x \left (a -1\right )\right ) a \right ) {\left (\frac {a y \left (x \right )-\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{x}\right )}^{\frac {1}{a -1}}+4 \left (-y \left (x \right ) \left (a -\frac {1}{2}\right ) \sqrt {a^{2} y \left (x \right )^{2}-4 b x}+\left (a^{2}-\frac {1}{2} a \right ) y \left (x \right )^{2}-2 b x \right ) x a}{\left (2 a -1\right ) \left (a y \left (x \right )-\sqrt {a^{2} y \left (x \right )^{2}-4 b x}\right )^{2}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.712 (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*}