60.7.135 problem 1726 (book 6.135)

Internal problem ID [11724]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1726 (book 6.135)
Date solved : Tuesday, January 28, 2025 at 06:11:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Solution by Maple

Time used: 0.083 (sec). Leaf size: 107

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

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

✓ Solution by Mathematica

Time used: 0.235 (sec). Leaf size: 109

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

\[ \text {Solve}\left [\left \{x=\int \frac {\exp \left (-\int _1^{K[4]}\frac {K[3]-1}{K[3]^3+K[3]^2+K[3]+1}dK[3]-c_1\right )}{K[4]^3+K[4]^2+K[4]+1} \, dK[4]+c_2,y(x)=x-\exp \left (-\int _1^{K[4]}\frac {K[3]-1}{K[3]^3+K[3]^2+K[3]+1}dK[3]-c_1\right )\right \},\{y(x),K[4]\}\right ] \]