60.1.89 problem 89

Internal problem ID [10103]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 89
Date solved : Monday, January 27, 2025 at 06:27:38 PM
CAS classification : [_quadrature]

\begin{align*} x y^{\prime }-\sqrt {a^{2}-x^{2}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 51

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

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 42

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

\[ y(x)\to -a \text {arctanh}\left (\frac {\sqrt {a^2-x^2}}{a}\right )+\sqrt {a^2-x^2}+c_1 \]