7.5.66 problem 72

Internal problem ID [170]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.6 (substitution and exact equations). Problems at page 72
Problem number : 72
Date solved : Friday, February 07, 2025 at 08:00:15 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 91

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

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ y &= \frac {\left (c_1 +r +x \right ) \left (c_1 -r +x \right )}{\sqrt {-c_1^{2}-2 c_1 x +r^{2}-x^{2}}}+c_2 \\ y &= \frac {\left (c_1 +r +x \right ) \left (-c_1 +r -x \right )}{\sqrt {-c_1^{2}-2 c_1 x +r^{2}-x^{2}}}+c_2 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.800 (sec). Leaf size: 71

DSolve[r*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-i \sqrt {\left (-1+c_1{}^2\right ) r^2+2 c_1 r x+x^2} \\ y(x)\to i \sqrt {\left (-1+c_1{}^2\right ) r^2+2 c_1 r x+x^2}+c_2 \\ \end{align*}