35.7.9 problem 6

Internal problem ID [6191]
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
Date solved : Monday, January 27, 2025 at 01:47:13 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear]]

\begin{align*} k&=\frac {y^{\prime \prime }}{\left (1+y^{\prime }\right )^{{3}/{2}}} \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 19

dsolve(k=diff(y(x),x$2)*(1+ (diff(y(x),x)))^(-3/2),y(x), singsol=all)
 

\[ y = -x -\frac {4}{k^{2} \left (x +c_{1} \right )}+c_{2} \]

✓ Solution by Mathematica

Time used: 0.585 (sec). Leaf size: 75

DSolve[k==D[y[x],{x,2}]*(1+ (D[y[x],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*}