78.8.4 problem 4

Internal problem ID [18202]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Miscellaneous Problems for Chapter 2. Problems at page 99
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 11:38:05 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 51

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

\[ \frac {y^{2}+\sqrt {x^{2}+y^{2}}\, y+\left (\ln \left (y+\sqrt {x^{2}+y^{2}}\right )-c_{1} -3 \ln \left (x \right )\right ) x^{2}}{x^{2}} = 0 \]

✓ Solution by Mathematica

Time used: 0.260 (sec). Leaf size: 66

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

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