27.1.6 problem 6

Internal problem ID [4300]
Book : An introduction to the solution and applications of differential equations, J.W. Searl, 1966
Section : Chapter 4, Ex. 4.1
Problem number : 6
Date solved : Monday, January 27, 2025 at 08:51:29 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

With initial conditions

\begin{align*} y \left (1\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.335 (sec). Leaf size: 33

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

\[ y \left (x \right ) = \frac {x \left (2 \ln \left (x \right )+\operatorname {RootOf}\left (4+4 \ln \left (x \right )^{2}+4 \ln \left (x \right ) \textit {\_Z} +\textit {\_Z}^{2}-4 \,{\mathrm e}^{\textit {\_Z}}\right )\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.172 (sec). Leaf size: 28

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

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