62.1.3 problem Ex 3

Internal problem ID [12800]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 8. Exact differential equations. Page 11
Problem number : Ex 3
Date solved : Tuesday, January 28, 2025 at 04:21:18 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} \frac {1}{\sqrt {x^{2}+y^{2}}}+\left (\frac {1}{y}-\frac {x}{y \sqrt {x^{2}+y^{2}}}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.194 (sec). Leaf size: 18

dsolve(1/sqrt(x^2+y(x)^2)+ ( 1/y(x)-(x/(y(x)*sqrt(x^2+y(x)^2))))*diff(y(x),x)=0,y(x), singsol=all)
 

\[ -c_{1} +x +\sqrt {x^{2}+y^{2}} = 0 \]

✓ Solution by Mathematica

Time used: 0.476 (sec). Leaf size: 62

DSolve[1/Sqrt[x^2+y[x]^2]+ ( 1/y[x]-(x/(y[x]*Sqrt[x^2+y[x]^2])))*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}} \\ y(x)\to e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to \text {ComplexInfinity} \\ \end{align*}