69.1.35 problem 53

Internal problem ID [14188]
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
Date solved : Tuesday, January 28, 2025 at 06:20:01 AM
CAS classification : [[_homogeneous, `class A`], _exact, _dAlembert]

\begin{align*} \frac {x +y^{\prime } y}{\sqrt {x^{2}+y^{2}}}&=m \end{align*}

✓ Solution by Maple

Time used: 0.114 (sec). Leaf size: 178

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

\[ \int _{\textit {\_b}}^{x}\frac {m \sqrt {\textit {\_a}^{2}+y^{2}}-\textit {\_a}}{-m \sqrt {\textit {\_a}^{2}+y^{2}}\, \textit {\_a} +y^{2}+\textit {\_a}^{2}}d \textit {\_a} -\int _{}^{y}\frac {\left (-1+\left (m \sqrt {\textit {\_f}^{2}+x^{2}}\, x -x^{2}-\textit {\_f}^{2}\right ) \left (\int _{\textit {\_b}}^{x}-\frac {2 \left (-\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \textit {\_a} +m \left (\textit {\_a}^{2}+\frac {\textit {\_f}^{2}}{2}\right )\right )}{\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \left (m \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \textit {\_a} -\textit {\_a}^{2}-\textit {\_f}^{2}\right )^{2}}d \textit {\_a} \right )\right ) \textit {\_f}}{m \sqrt {\textit {\_f}^{2}+x^{2}}\, x -x^{2}-\textit {\_f}^{2}}d \textit {\_f} +c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 2.146 (sec). Leaf size: 103

DSolve[(x+y[x]*D[y[x],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*}