31.6.4 problem 4

Internal problem ID [5753]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 4
Date solved : Monday, January 27, 2025 at 01:12:46 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} {y^{\prime }}^{2}+\frac {2 x y^{\prime }}{y}-1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.161 (sec). Leaf size: 43

dsolve((diff(y(x),x))^2+2*x/y(x)*diff(y(x),x)-1=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.470 (sec). Leaf size: 126

DSolve[(D[y[x],x])^2+2*x/y[x]*D[y[x],x]-1==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*}