Internal problem ID [4339]
Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley.
2006
Section: Chapter 8, Ordinary differential equations. Section 7. Other second-Order equations. page
435
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear]]
Solve \begin {gather*} \boxed {k -\frac {y^{\prime \prime }}{\left (y^{\prime }+1\right )^{\frac {3}{2}}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.057 (sec). Leaf size: 19
dsolve(k=diff(y(x),x$2)*(1+ (diff(y(x),x)))^(-3/2),y(x), singsol=all)
\[ y \relax (x ) = -x -\frac {4}{k^{2} \left (c_{1}+x \right )}+c_{2} \]
✓ Solution by Mathematica
Time used: 0.481 (sec). Leaf size: 73
DSolve[k==y''[x]*(1+ (y'[x])^2)^(-3/2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 k-i \sqrt {(k x-1+c_1) (k x+1+c_1)}}{k} \\ y(x)\to \frac {i \sqrt {(k x-1+c_1) (k x+1+c_1)}+c_2 k}{k} \\ \end{align*}