Internal problem ID [10193]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
25. Equations solvable for \(y\). Page 52
Problem number: Ex 2.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {4 x \left (y^{\prime }\right )^{2}+2 y^{\prime } x -y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 51
dsolve(4*x*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {x}{4} \\ y \relax (x ) = \left (\frac {4 c_{1}}{x}+\frac {2 \sqrt {x c_{1}}}{x}\right ) x \\ y \relax (x ) = \left (\frac {4 c_{1}}{x}-\frac {2 \sqrt {x c_{1}}}{x}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.133 (sec). Leaf size: 72
DSolve[4*x*(y'[x])^2+2*x*y'[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} e^{2 c_1} \left (-2 \sqrt {x}+e^{2 c_1}\right ) \\ y(x)\to \frac {1}{4} e^{-4 c_1} \left (1+2 e^{2 c_1} \sqrt {x}\right ) \\ y(x)\to 0 \\ y(x)\to -\frac {x}{4} \\ \end{align*}