Internal problem ID [12132]
Book: DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR
PUBLISHERS, Moscow 1969.
Section: Chapter 8. Differential equations. Exercises page 595
Problem number: 53.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _exact, _dAlembert]
\[ \boxed {\frac {x +y y^{\prime }}{\sqrt {x^{2}+y^{2}}}=m} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 194
dsolve((x+y(x)*diff(y(x),x))/sqrt(x^2+y(x)^2)=m,y(x), singsol=all)
\[ \int _{\textit {\_b}}^{x}-\frac {m \sqrt {\textit {\_a}^{2}+y \left (x \right )^{2}}-\textit {\_a}}{m \sqrt {\textit {\_a}^{2}+y \left (x \right )^{2}}\, \textit {\_a} -\textit {\_a}^{2}-y \left (x \right )^{2}}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (\frac {\textit {\_f}}{m \sqrt {\textit {\_f}^{2}+x^{2}}\, x -x^{2}-\textit {\_f}^{2}}-\left (\int _{\textit {\_b}}^{x}\left (\frac {\left (m \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}-\textit {\_a} \right ) \left (\frac {m \textit {\_a} \textit {\_f}}{\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}}-2 \textit {\_f} \right )}{\left (m \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \textit {\_a} -\textit {\_a}^{2}-\textit {\_f}^{2}\right )^{2}}-\frac {m \textit {\_f}}{\left (m \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \textit {\_a} -\textit {\_a}^{2}-\textit {\_f}^{2}\right ) \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 2.496 (sec). Leaf size: 103
DSolve[(x+y[x]*y'[x])/Sqrt[x^2+y[x]^2]==m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {\left (m^2-1\right ) x^2-2 e^{c_1} m x+e^{2 c_1}} y(x)\to \sqrt {\left (m^2-1\right ) x^2-2 e^{c_1} m x+e^{2 c_1}} y(x)\to -\sqrt {\left (m^2-1\right ) x^2} y(x)\to \sqrt {\left (m^2-1\right ) x^2} \end{align*}