62.14.2 problem Ex 2

Internal problem ID [12883]
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
Date solved : Tuesday, January 28, 2025 at 04:31:46 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} 4 x {y^{\prime }}^{2}+2 x y^{\prime }-y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 35

dsolve(4*x*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
 

\begin{align*} y &= -\frac {x}{4} \\ y &= 4 c_{1} +2 \sqrt {c_{1} x} \\ y &= 4 c_{1} -2 \sqrt {c_{1} x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.147 (sec). Leaf size: 72

DSolve[4*x*(D[y[x],x])^2+2*x*D[y[x],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*}