2.6.4 Problem 4 (eq 50)
Internal
problem
ID
[19739]
Book
:
Elementary
Differential
Equations.
By
Thornton
C.
Fry.
D
Van
Nostrand.
NY.
First
Edition
(1929)
Section
:
Chapter
IV.
Methods
of
solution:
First
order
equations.
section
33.
Problems
at
page
91
Problem
number
:
4
(eq
50)
Date
solved
:
Wednesday, January 28, 2026 at 11:02:25 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
2.6.4.1 second order ode missing x
20.784 (sec)
\begin{align*}
\phi ^{\prime \prime }&=\frac {4 \pi n c}{\sqrt {v_{0}^{2}+\frac {2 e \left (\phi -V_{0} \right )}{m}}} \\
\end{align*}
Entering second order ode missing \(x\) solverThis is missing independent variable second order ode.
Solved by reduction of order by using substitution which makes the dependent variable \(\phi \) an
independent variable. Using \begin{align*} \phi ' &= p \end{align*}
Then
\begin{align*} \phi '' &= \frac {dp}{dx}\\ &= \frac {dp}{d\phi }\frac {d\phi }{dx}\\ &= p \frac {dp}{d\phi } \end{align*}
Hence the ode becomes
\begin{align*} p \left (\phi \right ) \left (\frac {d}{d \phi }p \left (\phi \right )\right ) = \frac {4 \pi n c}{\sqrt {v_{0}^{2}+\frac {2 e \left (\phi -V_{0} \right )}{m}}} \end{align*}
Which is now solved as first order ode for \(p(\phi )\).
Entering first order ode separable solverThe ode
\begin{equation}
p^{\prime } = \frac {4 \pi n c}{\sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, p}
\end{equation}
is separable as it can be written as
\begin{align*} p^{\prime }&= \frac {4 \pi n c}{\sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, p}\\ &= f(\phi ) g(p) \end{align*}
Where
\begin{align*} f(\phi ) &= \frac {4 \pi n c}{\sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}}\\ g(p) &= \frac {1}{p} \end{align*}
Integrating gives
\begin{align*}
\int { \frac {1}{g(p)} \,dp} &= \int { f(\phi ) \,d\phi } \\
\int { p\,dp} &= \int { \frac {4 \pi n c}{\sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}} \,d\phi } \\
\end{align*}
\[
\frac {p^{2}}{2}=\frac {8 \pi n \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) c}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}+c_1
\]
Solving for \(p\) gives \begin{align*}
p &= \frac {\sqrt {2}\, \sqrt {e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (4 \pi c m n \,v_{0}^{2}-8 \pi V_{0} c e n +8 \pi c e n \phi +c_1 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\right )}}{e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}} \\
p &= -\frac {\sqrt {2}\, \sqrt {e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (4 \pi c m n \,v_{0}^{2}-8 \pi V_{0} c e n +8 \pi c e n \phi +c_1 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\right )}}{e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}} \\
\end{align*}
For solution (1) found earlier, since \(p=\phi ^{\prime }\) then the new first order
ode to solve is \begin{align*} \phi ^{\prime } = \frac {\sqrt {2}\, \sqrt {e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (4 \pi c m n \,v_{0}^{2}-8 \pi V_{0} c e n +8 \pi c e n \phi +c_1 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right )}}{e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}} \end{align*}
Entering first order ode autonomous solverIntegrating gives
\begin{align*} \int \frac {e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \sqrt {2}}{2 \sqrt {e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (4 \pi c m n \,v_{0}^{2}-8 \pi V_{0} c e n +8 \pi c e n \phi +c_1 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\right )}}d \phi &= dx\\ -\frac {\sqrt {2}\, \left (-4 \pi c m n \,v_{0}^{2}+8 \pi V_{0} c e n -8 \pi c e n \phi -c_1 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\right ) \sqrt {\frac {\left (-v_{0}^{2} m +2 e V_{0} -2 e \phi \right ) \left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \pi n c m +c_1 e \right )}{m}}\, \left (2 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \pi n c m -c_1 e \right )}{24 \sqrt {-e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (-4 \pi c m n \,v_{0}^{2}+8 \pi V_{0} c e n -8 \pi c e n \phi -c_1 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\right )}\, \sqrt {\sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (-4 \pi c m n \,v_{0}^{2}+8 \pi V_{0} c e n -8 \pi c e n \phi -c_1 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\right )}\, m \,\pi ^{2} n^{2} c^{2}}&= x +c_2 \end{align*}
Simplifying the above gives
\begin{align*}
\frac {\left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right ) \left (\pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -\frac {c_1 e}{2}\right ) \sqrt {-\frac {\left (v_{0}^{2} m +2 e \phi -2 e V_{0} \right ) \left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +c_1 e \right )}{m}}}{3 \sqrt {\sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, e \left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right )}\, \sqrt {-8 \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, \left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right )}\, m \,\pi ^{2} n^{2} c^{2}} &= x +c_2 \\
\end{align*}
For solution (2) found earlier, since \(p=\phi ^{\prime }\) then the new first order ode to
solve is \begin{align*} \phi ^{\prime } = -\frac {\sqrt {2}\, \sqrt {e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (4 \pi c m n \,v_{0}^{2}-8 \pi V_{0} c e n +8 \pi c e n \phi +c_1 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right )}}{e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}} \end{align*}
Entering first order ode autonomous solverIntegrating gives
\begin{align*} \int -\frac {e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \sqrt {2}}{2 \sqrt {e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (4 \pi c m n \,v_{0}^{2}-8 \pi V_{0} c e n +8 \pi c e n \phi +c_1 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\right )}}d \phi &= dx\\ \frac {\sqrt {2}\, \left (-4 \pi c m n \,v_{0}^{2}+8 \pi V_{0} c e n -8 \pi c e n \phi -c_1 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\right ) \sqrt {\frac {\left (-v_{0}^{2} m +2 e V_{0} -2 e \phi \right ) \left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \pi n c m +c_1 e \right )}{m}}\, \left (2 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \pi n c m -c_1 e \right )}{24 \sqrt {-e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (-4 \pi c m n \,v_{0}^{2}+8 \pi V_{0} c e n -8 \pi c e n \phi -c_1 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\right )}\, \sqrt {\sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (-4 \pi c m n \,v_{0}^{2}+8 \pi V_{0} c e n -8 \pi c e n \phi -c_1 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\right )}\, m \,\pi ^{2} n^{2} c^{2}}&= x +c_3 \end{align*}
Simplifying the above gives
\begin{align*}
-\frac {\left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right ) \left (\pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -\frac {c_1 e}{2}\right ) \sqrt {-\frac {\left (v_{0}^{2} m +2 e \phi -2 e V_{0} \right ) \left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +c_1 e \right )}{m}}}{3 \sqrt {\sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, e \left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right )}\, \sqrt {-8 \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, \left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right )}\, m \,\pi ^{2} n^{2} c^{2}} &= x +c_3 \\
\end{align*}
Summary of solutions found
\begin{align*}
-\frac {\left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right ) \left (\pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -\frac {c_1 e}{2}\right ) \sqrt {-\frac {\left (v_{0}^{2} m +2 e \phi -2 e V_{0} \right ) \left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +c_1 e \right )}{m}}}{3 \sqrt {\sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, e \left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right )}\, \sqrt {-8 \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, \left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right )}\, m \,\pi ^{2} n^{2} c^{2}} &= x +c_3 \\
\frac {\left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right ) \left (\pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -\frac {c_1 e}{2}\right ) \sqrt {-\frac {\left (v_{0}^{2} m +2 e \phi -2 e V_{0} \right ) \left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +c_1 e \right )}{m}}}{3 \sqrt {\sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, e \left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right )}\, \sqrt {-8 \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, \left (\frac {c_1 e \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}{8}+\pi c n \left (\frac {v_{0}^{2} m}{2}+e \phi -e V_{0} \right )\right )}\, m \,\pi ^{2} n^{2} c^{2}} &= x +c_2 \\
\end{align*}
2.6.4.2 second order ode can be made integrable
62.149 (sec)
\begin{align*}
\phi ^{\prime \prime }&=\frac {4 \pi n c}{\sqrt {v_{0}^{2}+\frac {2 e \left (\phi -V_{0} \right )}{m}}} \\
\end{align*}
Entering second order ode can be made integrable solverMultiplying the ode by \(\phi ^{\prime }\) gives \[ \phi ^{\prime } \phi ^{\prime \prime }-\frac {4 \phi ^{\prime } \pi n c}{\sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}} = 0 \]
Integrating the above w.r.t \(x\) gives \begin{align*} \int \left (\phi ^{\prime } \phi ^{\prime \prime }-\frac {4 \phi ^{\prime } \pi n c}{\sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}}\right )d x &= 0 \\ \frac {{\phi ^{\prime }}^{2}}{2}-\frac {4 \pi n c \sqrt {\frac {2 e \phi }{m}+\frac {v_{0}^{2} m -2 e V_{0}}{m}}\, m}{e} &= c_1 \end{align*}
Which is now solved for \(\phi \). Solving for the derivative gives these ODE’s to solve
\begin{align*}
\tag{1} \phi ^{\prime }&=\frac {\sqrt {2}\, \sqrt {e \left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +c_1 e \right )}}{e} \\
\tag{2} \phi ^{\prime }&=-\frac {\sqrt {2}\, \sqrt {e \left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +c_1 e \right )}}{e} \\
\end{align*}
Now each of the
above is solved separately.
Solving Eq. (1)
Entering first order ode autonomous solverIntegrating gives
\begin{align*} \int \frac {e \sqrt {2}}{2 \sqrt {e \left (4 \pi n c \sqrt {\frac {v_{0}^{2} m -2 e V_{0} +2 e \phi }{m}}\, m +c_1 e \right )}}d \phi &= dx\\ \frac {\sqrt {2}\, \left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \pi n c m +c_1 e \right ) \left (2 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \pi n c m -c_1 e \right )}{24 \sqrt {e \left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \pi n c m +c_1 e \right )}\, m \,c^{2} n^{2} \pi ^{2}}&= x +c_4 \end{align*}
Simplifying the above gives
\begin{align*}
\frac {\left (\pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +\frac {c_1 e}{4}\right ) \left (\pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -\frac {c_1 e}{2}\right ) \sqrt {2}}{3 \sqrt {4 e c m n \pi \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}+c_1 \,e^{2}}\, c^{2} m \,n^{2} \pi ^{2}} &= x +c_4 \\
\end{align*}
Solving Eq. (2)
Entering first order ode autonomous solverIntegrating gives
\begin{align*} \int -\frac {e \sqrt {2}}{2 \sqrt {e \left (4 \pi n c \sqrt {\frac {v_{0}^{2} m -2 e V_{0} +2 e \phi }{m}}\, m +c_1 e \right )}}d \phi &= dx\\ -\frac {\sqrt {2}\, \left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \pi n c m +c_1 e \right ) \left (2 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \pi n c m -c_1 e \right )}{24 \sqrt {e \left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \pi n c m +c_1 e \right )}\, m \,c^{2} n^{2} \pi ^{2}}&= x +c_5 \end{align*}
Simplifying the above gives
\begin{align*}
-\frac {\left (\pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +\frac {c_1 e}{4}\right ) \left (\pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -\frac {c_1 e}{2}\right ) \sqrt {2}}{3 \sqrt {4 e c m n \pi \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}+c_1 \,e^{2}}\, c^{2} m \,n^{2} \pi ^{2}} &= x +c_5 \\
\end{align*}
Summary of solutions found
\begin{align*}
\frac {\left (\pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +\frac {c_1 e}{4}\right ) \left (\pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -\frac {c_1 e}{2}\right ) \sqrt {2}}{3 \sqrt {4 e c m n \pi \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}+c_1 \,e^{2}}\, c^{2} m \,n^{2} \pi ^{2}} &= x +c_4 \\
\end{align*}
2.6.4.3 ✓ Maple. Time used: 0.064 (sec). Leaf size: 210
ode:=diff(diff(phi(x),x),x) = 4*Pi*n*c/(v__0^2+2*e/m*(phi(x)-V__0))^(1/2);
dsolve(ode,phi(x), singsol=all);
\begin{align*}
e \int _{}^{\phi }\frac {\sqrt {\frac {\left (-2 V_{0} +2 \textit {\_a} \right ) e +v_{0}^{2} m}{m}}}{4 \sqrt {\left (\frac {c_1 \sqrt {\left (2 V_{0} -2 \textit {\_a} \right ) e -v_{0}^{2} m}}{16}+\pi n c \left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right )\right ) e \sqrt {\frac {\left (-2 V_{0} +2 \textit {\_a} \right ) e +v_{0}^{2} m}{m}}}}d \textit {\_a} -x -c_2 &= 0 \\
-e \int _{}^{\phi }\frac {\sqrt {\frac {\left (-2 V_{0} +2 \textit {\_a} \right ) e +v_{0}^{2} m}{m}}}{4 \sqrt {\left (\frac {c_1 \sqrt {\left (2 V_{0} -2 \textit {\_a} \right ) e -v_{0}^{2} m}}{16}+\pi n c \left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right )\right ) e \sqrt {\frac {\left (-2 V_{0} +2 \textit {\_a} \right ) e +v_{0}^{2} m}{m}}}}d \textit {\_a} -x -c_2 &= 0 \\
\end{align*}
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
-> Calling odsolve with the ODE, diff(_b(_a),_a)*_b(_a)-4*Pi*n*c/(-(-m*v__0^2+2
*V__0*e-2*_a*e)/m)^(1/2) = 0, _b(_a), HINT = [[-2/3*(-m*v__0^2+2*V__0*e-2*_a*e)
/e, 1/3*_b]]
*** Sublevel 2 ***
symmetry methods on request
1st order, trying reduction of order with given symmetries:
[-2/3*(-m*v__0^2+2*V__0*e-2*_a*e)/e, 1/3*_b]
1st order, trying the canonical coordinates of the invariance group
-> Calling odsolve with the ODE, diff(y(x),x) = y(x)/((4*x-4*V__0)*e+2*
v__0^2*m)*e, y(x)
*** Sublevel 3 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
<- 1st order, canonical coordinates successful
<- differential order: 2; canonical coordinates successful
<- differential order 2; missing variables successful
2.6.4.4 ✓ Mathematica. Time used: 91.095 (sec). Leaf size: 2754
ode=D[phi[x],{x,2}]==4*Pi*n*c/Sqrt[v0^2+2*e/m*(phi[x]-V0)];
ic={};
DSolve[{ode,ic},phi[x],x,IncludeSingularSolutions->True]
Too large to display
2.6.4.5 ✗ Sympy
from sympy import *
x = symbols("x")
V__0 = symbols("V__0")
c = symbols("c")
e = symbols("e")
m = symbols("m")
n = symbols("n")
v__0 = symbols("v__0")
phi = Function("phi")
ode = Eq(-4*pi*c*n/sqrt(2*e*(-V__0 + phi(x))/m + v__0**2) + Derivative(phi(x), (x, 2)),0)
ics = {}
dsolve(ode,func=phi(x),ics=ics)
Timed Out
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0