Internal problem ID [3589]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 30
Problem number: 859.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}+y y^{\prime }-y^{4}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.296 (sec). Leaf size: 99
dsolve(x*diff(y(x),x)^2+y(x)*diff(y(x),x)-y(x)^4 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {1}{2 \sqrt {-x}} \\ y \relax (x ) = \frac {1}{2 \sqrt {-x}} \\ y \relax (x ) = 0 \\ y \relax (x ) = -\frac {\sqrt {-x \left (\tanh ^{2}\left (-\frac {\ln \relax (x )}{2}+\frac {c_{1}}{2}\right )-1\right )}}{2 x \tanh \left (-\frac {\ln \relax (x )}{2}+\frac {c_{1}}{2}\right )} \\ y \relax (x ) = \frac {\sqrt {-x \left (\tanh ^{2}\left (-\frac {\ln \relax (x )}{2}+\frac {c_{1}}{2}\right )-1\right )}}{2 x \tanh \left (-\frac {\ln \relax (x )}{2}+\frac {c_{1}}{2}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.527 (sec). Leaf size: 84
DSolve[x (y'[x])^2+y[x] y'[x]-y[x]^4==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 e^{\frac {c_1}{2}}}{-4 x+e^{c_1}} \\ y(x)\to \frac {2 e^{\frac {c_1}{2}}}{-4 x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to -\frac {i}{2 \sqrt {x}} \\ y(x)\to \frac {i}{2 \sqrt {x}} \\ \end{align*}