Internal problem ID [10156]
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]]
\[ \boxed {\left (y^{2} a^{2}-b^{2}\right ) {y^{\prime \prime }}^{2}-2 a^{2} y {y^{\prime }}^{2} y^{\prime \prime }+\left (a^{2} {y^{\prime }}^{2}-1\right ) {y^{\prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 124
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 {\operatorname {csgn}\left (\sec \left (\frac {c_{1} -x}{b}\right )\right ) \sin \left (\frac {c_{1} -x}{b}\right ) \operatorname {csgn}\left (a \right ) b}{a} \\ y \left (x \right ) &= -\frac {\operatorname {csgn}\left (\sec \left (\frac {c_{1} -x}{b}\right )\right ) \sin \left (\frac {c_{1} -x}{b}\right ) \operatorname {csgn}\left (a \right ) b}{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 (c_{2} +x \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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