60.7.177 problem 1768 (book 6.177)

Internal problem ID [11766]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1768 (book 6.177)
Date solved : Tuesday, January 28, 2025 at 06:11:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.246 (sec). Leaf size: 55

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

\begin{align*} y &= 0 \\ y &= c_{2} {\mathrm e}^{-\int \frac {x \sqrt {b^{2}-x^{2}}}{c_{1} \sqrt {b^{2}-x^{2}}+\left (b^{2}-x^{2}\right ) a}d x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 9.039 (sec). Leaf size: 54

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

\[ y(x)\to c_2 e^{\frac {\sqrt {b^2-x^2}}{a}} \left (a \sqrt {b^2-x^2}-c_1\right ){}^{\frac {c_1}{a^2}} \]