Internal problem ID [4149]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 8. Special second order equations. Lesson 35. Independent variable x
absent
Problem number: Exercise 35.7, page 504.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {r^{\prime \prime }+\frac {k}{r^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 369
dsolve(diff(r(t),t$2)=-k/(r(t)^2),r(t), singsol=all)
\begin{align*} r \relax (t ) = \frac {c_{1} \left (c_{1}^{2} k^{2}-2 k c_{1} {\mathrm e}^{\RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+{\mathrm e}^{2 \RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}\right ) {\mathrm e}^{-\RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}-2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}}{2} \\ r \relax (t ) = \frac {c_{1} \left (c_{1}^{2} k^{2}-2 k c_{1} {\mathrm e}^{\RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+{\mathrm e}^{2 \RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}\right ) {\mathrm e}^{-\RootOf \left (\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2}+2 \,\mathrm {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.167 (sec). Leaf size: 65
DSolve[r''[t]==-k/(r[t]^2),r[t],t,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left (\frac {r(t) \sqrt {\frac {2 k}{r(t)}+c_1}}{c_1}-\frac {2 k \tanh ^{-1}\left (\frac {\sqrt {\frac {2 k}{r(t)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}\right ){}^2=(t+c_2){}^2,r(t)\right ] \]