47.2.27 problem 27

Internal problem ID [7443]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 27
Date solved : Monday, January 27, 2025 at 02:59:50 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.269 (sec). Leaf size: 69

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

\begin{align*} y &= -i x \\ y &= i x \\ y &= 0 \\ y &= \sqrt {c_{1} \left (c_{1} -2 x \right )} \\ y &= \sqrt {c_{1} \left (c_{1} +2 x \right )} \\ y &= -\sqrt {c_{1} \left (c_{1} -2 x \right )} \\ y &= -\sqrt {c_{1} \left (c_{1} +2 x \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.476 (sec). Leaf size: 126

DSolve[y[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 -e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}} \\ y(x)\to e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}} \\ y(x)\to -e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}} \\ y(x)\to e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to -i x \\ y(x)\to i x \\ \end{align*}