Internal problem ID [4661]
Internal file name [OUTPUT/4154_Sunday_June_05_2022_12_30_08_PM_76444680/index.tex]
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 35.11, page 504.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_x"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {r^{\prime \prime }-\frac {h^{2}}{r^{3}}+\frac {k}{r^{2}}=0} \]
This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(r\) an independent variable. Using \begin {align*} r' &= p(r) \end {align*}
Then \begin {align*} r'' &= \frac {dp}{dt}\\ &= \frac {dr}{dt} \frac {dp}{dr}\\ &= p \frac {dp}{dr} \end {align*}
Hence the ode becomes \begin {align*} p \left (r \right ) \left (\frac {d}{d r}p \left (r \right )\right ) r^{3}+k r = h^{2} \end {align*}
Which is now solved as first order ode for \(p(r)\). In canonical form the ODE is \begin {align*} p' &= F(r,p)\\ &= f( r) g(p)\\ &= \frac {h^{2}-k r}{p \,r^{3}} \end {align*}
Where \(f(r)=\frac {h^{2}-k r}{r^{3}}\) and \(g(p)=\frac {1}{p}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{p}} \,dp &= \frac {h^{2}-k r}{r^{3}} \,d r \\ \int { \frac {1}{\frac {1}{p}} \,dp} &= \int {\frac {h^{2}-k r}{r^{3}} \,d r} \\ \frac {p^{2}}{2}&=-\frac {h^{2}}{2 r^{2}}+\frac {k}{r}+c_{1} \\ \end{align*} The solution is \[ \frac {p \left (r \right )^{2}}{2}+\frac {h^{2}}{2 r^{2}}-\frac {k}{r}-c_{1} = 0 \] For solution (1) found earlier, since \(p=r^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} \frac {{r^{\prime }}^{2}}{2}+\frac {h^{2}}{2 r^{2}}-\frac {k}{r}-c_{1} = 0 \end {align*}
Solving the given ode for \(r^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} r^{\prime }&=\frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{r} \tag {1} \\ r^{\prime }&=-\frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{r} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin{align*} \int \frac {r}{\sqrt {2 r^{2} c_{1} -h^{2}+2 k r}}d r &= \int d t \\ \frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}-\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}}&=t +c_{2} \\ \end{align*} Solving equation (2)
Integrating both sides gives \begin{align*} \int -\frac {r}{\sqrt {2 r^{2} c_{1} -h^{2}+2 k r}}d r &= \int d t \\ -\frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}+\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}}&=t +c_{3} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} \frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}-\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}} &= t +c_{2} \\ \tag{2} -\frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}+\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}} &= t +c_{3} \\ \end{align*}
Verification of solutions
\[ \frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}-\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}} = t +c_{2} \] Verified OK.
\[ -\frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}+\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}} = t +c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d t}r^{\prime }\right ) r^{3}+r k =h^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d t}r^{\prime } \\ \bullet & {} & \textrm {Define new dependent variable}\hspace {3pt} u \\ {} & {} & u \left (t \right )=r^{\prime } \\ \bullet & {} & \textrm {Compute}\hspace {3pt} \frac {d}{d t}r^{\prime } \\ {} & {} & u^{\prime }\left (t \right )=r^{\prime \prime } \\ \bullet & {} & \textrm {Use chain rule on the lhs}\hspace {3pt} \\ {} & {} & r^{\prime } \left (\frac {d}{d r}u \left (r \right )\right )=r^{\prime \prime } \\ \bullet & {} & \textrm {Substitute in the definition of}\hspace {3pt} u \\ {} & {} & u \left (r \right ) \left (\frac {d}{d r}u \left (r \right )\right )=r^{\prime \prime } \\ \bullet & {} & \textrm {Make substitutions}\hspace {3pt} r^{\prime }=u \left (r \right ),\frac {d}{d t}r^{\prime }=u \left (r \right ) \left (\frac {d}{d r}u \left (r \right )\right )\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & u \left (r \right ) \left (\frac {d}{d r}u \left (r \right )\right ) r^{3}+k r =h^{2} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d r}u \left (r \right )=\frac {h^{2}-k r}{u \left (r \right ) r^{3}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & u \left (r \right ) \left (\frac {d}{d r}u \left (r \right )\right )=\frac {h^{2}-k r}{r^{3}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} r \\ {} & {} & \int u \left (r \right ) \left (\frac {d}{d r}u \left (r \right )\right )d r =\int \frac {h^{2}-k r}{r^{3}}d r +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {u \left (r \right )^{2}}{2}=-\frac {h^{2}}{2 r^{2}}+\frac {k}{r}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (r \right ) \\ {} & {} & \left \{u \left (r \right )=\frac {\sqrt {2 r^{2} c_{1} -h^{2}+2 k r}}{r}, u \left (r \right )=-\frac {\sqrt {2 r^{2} c_{1} -h^{2}+2 k r}}{r}\right \} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (r \right ) \\ {} & {} & u \left (r \right )=\frac {\sqrt {2 r^{2} c_{1} -h^{2}+2 k r}}{r} \\ \bullet & {} & \textrm {Revert to original variables with substitution}\hspace {3pt} u \left (r \right )=r^{\prime },r =r \\ {} & {} & r^{\prime }=\frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{r} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & r^{\prime }=\frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{r} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {r^{\prime } r}{\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {r^{\prime } r}{\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}d t =\int 1d t +c_{2} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}-\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}}=t +c_{2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} r \\ {} & {} & \left \{\frac {\mathit {RootOf}\left (-4 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k \mathit {RootOf}\left (-8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}+32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} c_{2} k \textit {\_Z} -8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}+32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} k t \textit {\_Z} +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k +4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}-4 h^{2} c_{1}^{3}-2 c_{1}^{2} k^{2}\right )-4 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t \mathit {RootOf}\left (-8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}+32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} c_{2} k \textit {\_Z} -8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}+32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} k t \textit {\_Z} +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k +4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}-4 h^{2} c_{1}^{3}-2 c_{1}^{2} k^{2}\right )-8 c_{1}^{5} c_{2}^{2}-16 c_{1}^{5} c_{2} t -8 c_{1}^{5} t^{2}-c_{1}^{2} k^{2} {\mathit {RootOf}\left (-8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}+32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} c_{2} k \textit {\_Z} -8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}+32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} k t \textit {\_Z} +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k +4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}-4 h^{2} c_{1}^{3}-2 c_{1}^{2} k^{2}\right )}^{2}-2 h^{2} c_{1}^{3}+2 k c_{1} \textit {\_Z} +\textit {\_Z}^{2}\right )}{2 c_{1}^{2}}\right \} \\ \bullet & {} & \textrm {Solve 2nd ODE for}\hspace {3pt} u \left (r \right ) \\ {} & {} & u \left (r \right )=-\frac {\sqrt {2 r^{2} c_{1} -h^{2}+2 k r}}{r} \\ \bullet & {} & \textrm {Revert to original variables with substitution}\hspace {3pt} u \left (r \right )=r^{\prime },r =r \\ {} & {} & r^{\prime }=-\frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{r} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & r^{\prime }=-\frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{r} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {r^{\prime } r}{\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}=-1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {r^{\prime } r}{\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}d t =\int \left (-1\right )d t +c_{2} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}-\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}}=-t +c_{2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} r \\ {} & {} & \left \{\frac {\mathit {RootOf}\left (-4 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k \mathit {RootOf}\left (-8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}-32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} +16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} c_{2} k \textit {\_Z} +8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}-32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} +16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} k t \textit {\_Z} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} +16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}-4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 h^{2} c_{1}^{3}-2 c_{1}^{2} k^{2}\right )+4 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t \mathit {RootOf}\left (-8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}-32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} +16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} c_{2} k \textit {\_Z} +8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}-32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} +16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} k t \textit {\_Z} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} +16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}-4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 h^{2} c_{1}^{3}-2 c_{1}^{2} k^{2}\right )-8 c_{1}^{5} c_{2}^{2}+16 c_{1}^{5} c_{2} t -8 c_{1}^{5} t^{2}-c_{1}^{2} k^{2} {\mathit {RootOf}\left (-8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}-32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} +16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} c_{2} k \textit {\_Z} +8 \mathit {RootOf}\left (8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} k t -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} k +16 c_{1}^{5} c_{2}^{2}-32 c_{1}^{5} c_{2} t +16 c_{1}^{5} t^{2}-16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} +16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +2 c_{1}^{2} k^{2} \textit {\_Z}^{2}+4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}+4 c_{1}^{2} \textit {\_Z} \,k^{2}+2 c_{1}^{2} k^{2}-4, \mathit {index} =1\right ) c_{1}^{\frac {7}{2}} k t \textit {\_Z} -16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} c_{2} +16 c_{1}^{4} {\mathrm e}^{\textit {\_Z}} t +8 \sqrt {2}\, c_{1}^{\frac {7}{2}} c_{2} \textit {\_Z} k -8 \sqrt {2}\, c_{1}^{\frac {7}{2}} \textit {\_Z} k t +4 c_{1}^{3} \left ({\mathrm e}^{\textit {\_Z}}\right )^{2}-4 \sqrt {2}\, c_{1}^{\frac {5}{2}} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} k -4 h^{2} c_{1}^{3}-2 c_{1}^{2} k^{2}\right )}^{2}-2 h^{2} c_{1}^{3}+2 k c_{1} \textit {\_Z} +\textit {\_Z}^{2}\right )}{2 c_{1}^{2}}\right \} \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)+(_a*k-h^2)/_a^3 = 0, _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful <- differential order: 2; canonical coordinates successful <- differential order 2; missing variables successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 441
dsolve(diff(r(t),t$2)= h^2/r(t)^3-k/r(t)^2,r(t), singsol=all)
\begin{align*} r \left (t \right ) &= \frac {c_{1} \left (c_{1}^{2} k^{2}-2 k c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+h^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}}{2} \\ r \left (t \right ) &= \frac {c_{1} \left (c_{1}^{2} k^{2}-2 k c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+h^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.099 (sec). Leaf size: 130
DSolve[r''[t]==h^2/r[t]^3-k/r[t]^2,r[t],t,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\left (\sqrt {c_1} \left (-h^2+r(t) (2 k+c_1 r(t))\right )-k \sqrt {-h^2+r(t) (2 k+c_1 r(t))} \text {arctanh}\left (\frac {k+c_1 r(t)}{\sqrt {c_1} \sqrt {-h^2+r(t) (2 k+c_1 r(t))}}\right )\right ){}^2}{c_1{}^3 r(t)^2 \left (-\frac {h^2}{r(t)^2}+\frac {2 k}{r(t)}+c_1\right )}=(t+c_2){}^2,r(t)\right ] \]