Internal problem ID [8037]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 457.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {x^{4} \left (y^{\prime }\right )^{2}-x y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 135
dsolve(x^4*diff(y(x),x)^2-x*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {1}{4 x^{2}} \\ y \relax (x ) = \frac {-c_{1}^{2}-c_{1} \left (2 i x -c_{1}\right )-2 x^{2}}{2 x^{2} c_{1}^{2}} \\ y \relax (x ) = \frac {-c_{1}^{2}-c_{1} \left (-2 i x -c_{1}\right )-2 x^{2}}{2 x^{2} c_{1}^{2}} \\ y \relax (x ) = \frac {c_{1} \left (2 i x +c_{1}\right )-2 x^{2}-c_{1}^{2}}{2 c_{1}^{2} x^{2}} \\ y \relax (x ) = \frac {c_{1} \left (-2 i x +c_{1}\right )-2 x^{2}-c_{1}^{2}}{2 c_{1}^{2} x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.568 (sec). Leaf size: 123
DSolve[-y[x] - x*y'[x] + x^4*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {x \sqrt {4 x^2 y(x)+1} \tanh ^{-1}\left (\sqrt {4 x^2 y(x)+1}\right )}{\sqrt {4 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {x \sqrt {4 x^2 y(x)+1} \tanh ^{-1}\left (\sqrt {4 x^2 y(x)+1}\right )}{\sqrt {4 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}