Internal problem ID [4847]
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: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear]]
\[ \boxed {-\frac {y^{\prime \prime }}{\left (1+y^{\prime }\right )^{\frac {3}{2}}}=-k} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 19
dsolve(k=diff(y(x),x$2)*(1+ (diff(y(x),x)))^(-3/2),y(x), singsol=all)
\[ y \left (x \right ) = -x -\frac {4}{k^{2} \left (x +c_{1} \right )}+c_{2} \]
✓ Solution by Mathematica
Time used: 0.515 (sec). Leaf size: 75
DSolve[k==y''[x]*(1+ (y'[x])^2)^(-3/2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\frac {i \sqrt {k^2 x^2+2 c_1 k x-1+c_1{}^2}}{k} y(x)\to \frac {i \sqrt {k^2 x^2+2 c_1 k x-1+c_1{}^2}}{k}+c_2 \end{align*}