32.1.6 problem First order with homogeneous Coefficients. Exercise 7.7, page 61

Internal problem ID [5776]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 7
Problem number : First order with homogeneous Coefficients. Exercise 7.7, page 61
Date solved : Monday, January 27, 2025 at 01:16:56 PM
CAS classification : [[_homogeneous, `class G`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 32

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

\[ \frac {-c_{1} y \left (x \right ) x +\sqrt {y \left (x \right )^{2}-x^{2}}+y \left (x \right )}{y \left (x \right ) x} = 0 \]

✓ Solution by Mathematica

Time used: 3.668 (sec). Leaf size: 107

DSolve[y[x]^2+(x*Sqrt[y[x]^2-x^2]-x*y[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [2 \text {arctanh}\left (\sqrt {\frac {\frac {y(x)}{x}-1}{\frac {y(x)}{x}+1}}\right )-\frac {\sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {y(x)}{x}+1} \left (\log \left (\sqrt {\frac {y(x)}{x}+1}-1\right )+\log \left (\sqrt {\frac {y(x)}{x}+1}+1\right )\right )}{\sqrt {\frac {y(x)^2}{x^2}-1}}=\log (x)+c_1,y(x)\right ] \]