29.19.17 problem 530

Internal problem ID [5126]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 19
Problem number : 530
Date solved : Monday, January 27, 2025 at 10:13:14 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 48

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

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

✓ Solution by Mathematica

Time used: 0.343 (sec). Leaf size: 109

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

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