60.7.234 problem 1825 (book 6.234)

Internal problem ID [11823]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1825 (book 6.234)
Date solved : Tuesday, January 28, 2025 at 06:11:30 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.350 (sec). Leaf size: 122

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

\begin{align*} y &= -i x +c_{1} \\ y &= i x +c_{1} \\ y &= \frac {c_{2} a +\int \frac {-c_{1} a^{2}+x \sqrt {a^{2} \left (c_{1}^{2}+a^{2}-x^{2}\right )}}{a^{2}-x^{2}}d x}{a} \\ y &= \frac {c_{2} a -\int \frac {c_{1} a^{2}+x \sqrt {a^{2} \left (c_{1}^{2}+a^{2}-x^{2}\right )}}{a^{2}-x^{2}}d x}{a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.665 (sec). Leaf size: 331

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

\begin{align*} y(x)\to -\frac {\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )} \left (c_1 \arctan \left (\frac {a^2-a x+c_1{}^2}{c_1 \sqrt {-a^2+x^2-c_1{}^2}}\right )+c_1 \arctan \left (\frac {a^2+a x+c_1{}^2}{c_1 \sqrt {-a^2+x^2-c_1{}^2}}\right )+2 \sqrt {-a^2+x^2-c_1{}^2}\right )}{2 x \sqrt {-a^2+x^2-c_1{}^2}}+c_1 \left (-\text {arctanh}\left (\frac {x}{a}\right )\right )+c_2 \\ y(x)\to \frac {\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )} \left (c_1 \arctan \left (\frac {a^2-a x+c_1{}^2}{c_1 \sqrt {-a^2+x^2-c_1{}^2}}\right )+c_1 \arctan \left (\frac {a^2+a x+c_1{}^2}{c_1 \sqrt {-a^2+x^2-c_1{}^2}}\right )+2 \sqrt {-a^2+x^2-c_1{}^2}\right )}{2 x \sqrt {-a^2+x^2-c_1{}^2}}+c_1 \left (-\text {arctanh}\left (\frac {x}{a}\right )\right )+c_2 \\ \end{align*}