Internal problem ID [10165]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1834 (book 6.243).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {\left (a^{2} y^{2}-b^{2}\right ) {y^{\prime \prime }}^{2}-2 a^{2} y {y^{\prime }}^{2} y^{\prime \prime }+\left ({y^{\prime }}^{2} a^{2}-1\right ) {y^{\prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 162
dsolve((a^2*y(x)^2-b^2)*diff(diff(y(x),x),x)^2-2*a^2*y(x)*diff(y(x),x)^2*diff(diff(y(x),x),x)+(a^2*diff(y(x),x)^2-1)*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\tan \left (\frac {\sqrt {a^{2}}\, \left (-x +c_{1} \right )}{b a}\right ) b}{\sqrt {{\tan \left (\frac {\sqrt {a^{2}}\, \left (-x +c_{1} \right )}{b a}\right )}^{2}+1}\, a} y \left (x \right ) = -\frac {\tan \left (\frac {\sqrt {a^{2}}\, \left (-x +c_{1} \right )}{b a}\right ) b}{\sqrt {{\tan \left (\frac {\sqrt {a^{2}}\, \left (-x +c_{1} \right )}{b a}\right )}^{2}+1}\, a} y \left (x \right ) = -\frac {b}{a} y \left (x \right ) = \frac {b}{a} y \left (x \right ) = c_{1} y \left (x \right ) = \frac {b \left ({\mathrm e}^{\frac {\sqrt {c_{1}^{2} a^{2}-1}\, \left (x +c_{2} \right )}{b}}-c_{1} \right )}{\sqrt {c_{1}^{2} a^{2}-1}} \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]^2*(-1 + a^2*y'[x]^2) - 2*a^2*y[x]*y'[x]^2*y''[x] + (-b^2 + a^2*y[x]^2)*y''[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
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