77.1.58 problem 77 (page 120)

Internal problem ID [17948]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 77 (page 120)
Date solved : Tuesday, January 28, 2025 at 11:14:52 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 1.462 (sec). Leaf size: 378

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

\begin{align*} -{\mathrm e}^{k \left (\int _{}^{\frac {-k^{2} x +\sqrt {\left (k^{2}-1\right ) y^{2}+x^{2} k^{2}}}{\left (k^{2}-1\right ) y}}\frac {k \sqrt {\textit {\_a}^{2}+1}-\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}\, \left (-\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} +k \left (\textit {\_a}^{2}+1\right )\right ) \left (k \textit {\_a} -\sqrt {\textit {\_a}^{2}+1}\right )}d \textit {\_a} \right )} c_{1} +x &= 0 \\ -{\mathrm e}^{k \left (\int _{}^{\frac {-k^{2} x -\sqrt {\left (k^{2}-1\right ) y^{2}+x^{2} k^{2}}}{y \left (k^{2}-1\right )}}\frac {k \sqrt {\textit {\_a}^{2}+1}-\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}\, \left (-\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} +k \left (\textit {\_a}^{2}+1\right )\right ) \left (k \textit {\_a} -\sqrt {\textit {\_a}^{2}+1}\right )}d \textit {\_a} \right )} c_{1} +x &= 0 \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\left (\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}-\sqrt {\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}}\right ) \textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}\right )}d \textit {\_a} +c_{1} \right ) x \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}+\sqrt {\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}}\right ) \textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}\right )}d \textit {\_a} +c_{1} \right ) x \\ \end{align*}

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

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

Timed out