Internal problem ID [3558]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 827.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+2 x y^{3} y^{\prime }+y^{4}=0} \end {gather*}
✓ Solution by Maple
Time used: 2.188 (sec). Leaf size: 51
dsolve(diff(y(x),x)^2+2*x*y(x)^3*diff(y(x),x)+y(x)^4 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {1}{x} \\ y \relax (x ) = \frac {1}{x} \\ y \relax (x ) = 0 \\ y \relax (x ) = \frac {1}{\sqrt {2 c_{1} x -c_{1}^{2}}} \\ y \relax (x ) = -\frac {1}{\sqrt {2 c_{1} x -c_{1}^{2}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.962 (sec). Leaf size: 161
DSolve[(y'[x])^2+2 x y[x]^3 y'[x]+y[x]^4==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {\sqrt {x^2 y(x)^2-1} y(x)^2 \tanh ^{-1}\left (\frac {x y(x)}{\sqrt {x^2 y(x)^2-1}}\right )}{\sqrt {y(x)^4 \left (x^2 y(x)^2-1\right )}}-\log (y(x))=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {y(x)^2 \sqrt {x^2 y(x)^2-1} \tanh ^{-1}\left (\frac {x y(x)}{\sqrt {x^2 y(x)^2-1}}\right )}{\sqrt {y(x)^4 \left (x^2 y(x)^2-1\right )}}-\log (y(x))=c_1,y(x)\right ] \\ y(x)\to 0 \\ y(x)\to -\frac {1}{x} \\ y(x)\to \frac {1}{x} \\ \end{align*}