15.19.4 problem 4

Internal problem ID [3288]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 4
Date solved : Monday, January 27, 2025 at 07:31:08 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x \left ({y^{\prime }}^{2}-1\right )&=2 y y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.076 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.084 (sec). Leaf size: 45

DSolve[x*(D[y[x],x]^2-1)==2*y[x]*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to x \sinh (-\log (x)+c_1) \\ y(x)\to x \sinh (\log (x)+c_1) \\ y(x)\to -i x \\ y(x)\to i x \\ \end{align*}