26.5.3 problem 4

Internal problem ID [4277]
Book : Differential equations with applications and historial notes, George F. Simmons. Second edition. 1971
Section : Chapter 2, End of chapter, page 61
Problem number : 4
Date solved : Monday, January 27, 2025 at 08:47:28 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: 0.007 (sec). Leaf size: 51

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

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

✓ Solution by Mathematica

Time used: 0.295 (sec). Leaf size: 66

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

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