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 : Tuesday, March 10, 2026 at 12:05:42 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

2.6.4.1 second order ode missing x

65.680 (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*}

\[ \frac {p^{2}}{2}=\frac {8 n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}+c_1 \]

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 exact solver To solve an ode of the form

\begin{equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0\tag {A}\end{equation}

We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives

\[ \frac {d}{dx}\phi \left ( x,y\right ) =0 \]

Hence

\begin{equation} \frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B}\end{equation}

Comparing (A,B) shows that

\begin{align*} \frac {\partial \phi }{\partial x} & =M\\ \frac {\partial \phi }{\partial y} & =N \end{align*}

But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that

\[ \frac {\partial M}{\partial y}=\frac {\partial N}{\partial x}\]

If the above condition is satisfied, then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is

\[ M(x,\phi ) \mathop {\mathrm {d}x}+ N(x,\phi ) \mathop {\mathrm {d}\phi }=0 \tag {1A} \]

Therefore

\begin{align*} \mathop {\mathrm {d}\phi } &= \left (\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}}}\right )\mathop {\mathrm {d}x}\\ \left (-\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}}}\right ) \mathop {\mathrm {d}x} + \mathop {\mathrm {d}\phi } &= 0 \tag {2A} \end{align*}

Comparing (1A) and (2A) shows that

\begin{align*} M(x,\phi ) &= -\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}}}\\ N(x,\phi ) &= 1 \end{align*}

The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied

\[ \frac {\partial M}{\partial \phi } = \frac {\partial N}{\partial x} \]

Using result found above gives

\begin{align*} \frac {\partial M}{\partial \phi } &= \frac {\partial }{\partial \phi } \left (-\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}}}\right )\\ &= -\frac {c n \pi \sqrt {2}\, e \sqrt {8}}{4 \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}} \end{align*}

And

\begin{align*} \frac {\partial N}{\partial x} &= \frac {\partial }{\partial x} \left (1\right )\\ &= 0 \end{align*}

Since \(\frac {\partial M}{\partial \phi } \neq \frac {\partial N}{\partial x}\), then the ODE is not exact. Since the ODE is not exact, we will try to find an integrating factor to make it exact. Let

\begin{align*} A &= \frac {1}{N} \left (\frac {\partial M}{\partial \phi } - \frac {\partial N}{\partial x} \right ) \\ &=1\left ( \left ( \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 )}}{\left (-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}\right )^{{3}/{2}} m}-\frac {\sqrt {2}\, \left (\frac {e^{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 {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, m}+e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (8 \pi c n e +\frac {c_1 \,e^{2}}{\sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, m}\right )\right )}{2 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \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 )}}\right ) - \left (0 \right ) \right ) \\ &=-\frac {c n \pi \sqrt {2}\, e \sqrt {8}}{4 \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}} \end{align*}

Since \(A\) depends on \(\phi \), it can not be used to obtain an integrating factor. We will now try a second method to find an integrating factor. Let

\begin{align*} B &= \frac {1}{M} \left ( \frac {\partial N}{\partial x} - \frac {\partial M}{\partial \phi } \right ) \\ &=-\frac {e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{4 \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}}\left ( \left ( 0\right ) - \left (\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 )}}{\left (-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}\right )^{{3}/{2}} m}-\frac {\sqrt {2}\, \left (\frac {e^{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 {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, m}+e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (8 \pi c n e +\frac {c_1 \,e^{2}}{\sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, m}\right )\right )}{2 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \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 )}} \right ) \right ) \\ &=-\frac {2 \pi c e n}{c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}+8 n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )} \end{align*}

Since \(B\) does not depend on \(x\), it can be used to obtain an integrating factor. Let the integrating factor be \(\mu \). Then

\begin{align*} \mu &= e^{\int B \mathop {\mathrm {d}\phi }} \\ &= e^{\int -\frac {2 \pi c e n}{c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}+8 n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )}\mathop {\mathrm {d}\phi } } \end{align*}

The result of integrating gives

\begin{align*} \mu &= e^{\frac {\ln \left (4 \pi \sqrt {\frac {2 e \phi }{m}-\frac {-v_{0}^{2} m +2 e V_{0}}{m}}\, c m n -c_1 e \right )}{4}-\frac {\ln \left (4 \pi \sqrt {\frac {2 e \phi }{m}-\frac {-v_{0}^{2} m +2 e V_{0}}{m}}\, c m n +c_1 e \right )}{4}-\frac {\ln \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 c^{2} \phi e m \,n^{2} \pi ^{2}+c_1^{2} e^{2}\right )}{4} } \\ &= \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}} \end{align*}

\(M\) and \(N\) are now multiplied by this integrating factor, giving new \(M\) and new \(N\) which are called \(\overline {M}\) and \(\overline {N}\) so not to confuse them with the original \(M\) and \(N\).

\begin{align*} \overline {M} &=\mu M \\ &= \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}\left (-\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}}}\right ) \\ &= -\frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e} \end{align*}

And

\begin{align*} \overline {N} &=\mu N \\ &= \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}\left (1\right ) \\ &= \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}} \end{align*}

So now a modified ODE is obtained from the original ODE which will be exact and can be solved using the standard method. The modified ODE is

\begin{align*} \overline {M} + \overline {N} \frac { \mathop {\mathrm {d}\phi }}{\mathop {\mathrm {d}x}} &= 0 \\ \left (-\frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e}\right ) + \left (\frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}\right ) \frac { \mathop {\mathrm {d}\phi }}{\mathop {\mathrm {d}x}} &= 0 \end{align*}

The following equations are now set up to solve for the function \(\phi \left (x,\phi \right )\)

\begin{align*} \frac {\partial \phi }{\partial x } &= \overline {M}\tag {1} \\ \frac {\partial \phi }{\partial \phi } &= \overline {N}\tag {2} \end{align*}

Integrating (1) w.r.t. \(x\) gives

\begin{align*} \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int \overline {M}\mathop {\mathrm {d}x} \\ \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int -\frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e}\mathop {\mathrm {d}x} \\ \tag{3} \phi &= -\frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}} x}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e}+ f(\phi ) \\ \end{align*}

Where \(f(\phi )\) is used for the constant of integration since \(\phi \) is a function of both \(x\) and \(\phi \). Taking derivative of equation (3) w.r.t \(\phi \) gives

\begin{align*} \tag{4} \frac {\partial \phi }{\partial \phi } &= \text {Expression too large to display}+f'(\phi ) \\ &=0+f'(\phi ) \\ \end{align*}

But equation (2) says that \(\frac {\partial \phi }{\partial \phi } = \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}\). Therefore equation (4) becomes

\begin{equation} \tag{5} \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}} = 0+f'(\phi ) \end{equation}

Solving equation (5) for \( f'(\phi )\) gives

\[ f'(\phi ) = \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}} \]

Integrating the above w.r.t \(\phi \) gives

\begin{align*} \int f'(\phi ) \mathop {\mathrm {d}\phi } &= \int \left ( \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}\right ) \mathop {\mathrm {d}\phi } \\ f(\phi ) &= \int \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \phi + c_3 \\ \end{align*}

\[ \phi = -\frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}} x}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e}+\int \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \phi + c_3 \]

But since \(\phi \) itself is a constant function, then let \(\phi =c_4\) where \(c_2\) is new constant and combining \(c_3\) and \(c_4\) constants into the constant \(c_3\) gives the solution as

\[ c_3 = -\frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}} x}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e}+\int \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \phi \]

Simplifying the above gives

\begin{align*} -\frac {-\int _{}^{\phi }\frac {{\left (4 \sqrt {\frac {2 e \left (\textit {\_a} -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\textit {\_a} -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\textit {\_a} -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \textit {\_a} \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, {\left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-32 c^{2} \phi e m \,n^{2} \pi ^{2}+32 n^{2} c^{2} \left (-\frac {v_{0}^{2} m}{2}+e V_{0} \right ) m \,\pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e +\sqrt {2}\, \sqrt {8}\, \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 )}\, {\left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -c_1 e \right )}^{{1}/{4}} x}{\sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, {\left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-32 c^{2} \phi e m \,n^{2} \pi ^{2}+32 n^{2} c^{2} \left (-\frac {v_{0}^{2} m}{2}+e V_{0} \right ) m \,\pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e} &= c_3 \\ \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 exact solver To solve an ode of the form

\begin{equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0\tag {A}\end{equation}

We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives

\[ \frac {d}{dx}\phi \left ( x,y\right ) =0 \]

Hence

\begin{equation} \frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B}\end{equation}

Comparing (A,B) shows that

\begin{align*} \frac {\partial \phi }{\partial x} & =M\\ \frac {\partial \phi }{\partial y} & =N \end{align*}

But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that

\[ \frac {\partial M}{\partial y}=\frac {\partial N}{\partial x}\]

If the above condition is satisfied, then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is

\[ M(x,\phi ) \mathop {\mathrm {d}x}+ N(x,\phi ) \mathop {\mathrm {d}\phi }=0 \tag {1A} \]

Therefore

\begin{align*} \mathop {\mathrm {d}\phi } &= \left (-\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}}}\right )\mathop {\mathrm {d}x}\\ \left (\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}}}\right ) \mathop {\mathrm {d}x} + \mathop {\mathrm {d}\phi } &= 0 \tag {2A} \end{align*}

Comparing (1A) and (2A) shows that

\begin{align*} M(x,\phi ) &= \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}}}\\ N(x,\phi ) &= 1 \end{align*}

The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied

\[ \frac {\partial M}{\partial \phi } = \frac {\partial N}{\partial x} \]

Using result found above gives

\begin{align*} \frac {\partial M}{\partial \phi } &= \frac {\partial }{\partial \phi } \left (\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}}}\right )\\ &= \frac {c n \pi \sqrt {2}\, e \sqrt {8}}{4 \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}} \end{align*}

And

\begin{align*} \frac {\partial N}{\partial x} &= \frac {\partial }{\partial x} \left (1\right )\\ &= 0 \end{align*}

Since \(\frac {\partial M}{\partial \phi } \neq \frac {\partial N}{\partial x}\), then the ODE is not exact. Since the ODE is not exact, we will try to find an integrating factor to make it exact. Let

\begin{align*} A &= \frac {1}{N} \left (\frac {\partial M}{\partial \phi } - \frac {\partial N}{\partial x} \right ) \\ &=1\left ( \left ( -\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 )}}{\left (-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}\right )^{{3}/{2}} m}+\frac {\sqrt {2}\, \left (\frac {e^{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 {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, m}+e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (8 \pi c n e +\frac {c_1 \,e^{2}}{\sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, m}\right )\right )}{2 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \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 )}}\right ) - \left (0 \right ) \right ) \\ &=\frac {c n \pi \sqrt {2}\, e \sqrt {8}}{4 \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}} \end{align*}

Since \(A\) depends on \(\phi \), it can not be used to obtain an integrating factor. We will now try a second method to find an integrating factor. Let

\begin{align*} B &= \frac {1}{M} \left ( \frac {\partial N}{\partial x} - \frac {\partial M}{\partial \phi } \right ) \\ &=\frac {e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{4 \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}}\left ( \left ( 0\right ) - \left (-\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 )}}{\left (-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}\right )^{{3}/{2}} m}+\frac {\sqrt {2}\, \left (\frac {e^{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 {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, m}+e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (8 \pi c n e +\frac {c_1 \,e^{2}}{\sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, m}\right )\right )}{2 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \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 )}} \right ) \right ) \\ &=-\frac {2 \pi c e n}{c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}+8 n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )} \end{align*}

Since \(B\) does not depend on \(x\), it can be used to obtain an integrating factor. Let the integrating factor be \(\mu \). Then

\begin{align*} \mu &= e^{\int B \mathop {\mathrm {d}\phi }} \\ &= e^{\int -\frac {2 \pi c e n}{c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}+8 n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )}\mathop {\mathrm {d}\phi } } \end{align*}

The result of integrating gives

\begin{align*} \mu &= e^{\frac {\ln \left (4 \pi \sqrt {\frac {2 e \phi }{m}-\frac {-v_{0}^{2} m +2 e V_{0}}{m}}\, c m n -c_1 e \right )}{4}-\frac {\ln \left (4 \pi \sqrt {\frac {2 e \phi }{m}-\frac {-v_{0}^{2} m +2 e V_{0}}{m}}\, c m n +c_1 e \right )}{4}-\frac {\ln \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 c^{2} \phi e m \,n^{2} \pi ^{2}+c_1^{2} e^{2}\right )}{4} } \\ &= \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}} \end{align*}

\(M\) and \(N\) are now multiplied by this integrating factor, giving new \(M\) and new \(N\) which are called \(\overline {M}\) and \(\overline {N}\) so not to confuse them with the original \(M\) and \(N\).

\begin{align*} \overline {M} &=\mu M \\ &= \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}\left (\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}}}\right ) \\ &= \frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e} \end{align*}

And

\begin{align*} \overline {N} &=\mu N \\ &= \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}\left (1\right ) \\ &= \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}} \end{align*}

So now a modified ODE is obtained from the original ODE which will be exact and can be solved using the standard method. The modified ODE is

\begin{align*} \overline {M} + \overline {N} \frac { \mathop {\mathrm {d}\phi }}{\mathop {\mathrm {d}x}} &= 0 \\ \left (\frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e}\right ) + \left (\frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}\right ) \frac { \mathop {\mathrm {d}\phi }}{\mathop {\mathrm {d}x}} &= 0 \end{align*}

The following equations are now set up to solve for the function \(\phi \left (x,\phi \right )\)

\begin{align*} \frac {\partial \phi }{\partial x } &= \overline {M}\tag {1} \\ \frac {\partial \phi }{\partial \phi } &= \overline {N}\tag {2} \end{align*}

Integrating (1) w.r.t. \(x\) gives

\begin{align*} \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int \overline {M}\mathop {\mathrm {d}x} \\ \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int \frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e}\mathop {\mathrm {d}x} \\ \tag{3} \phi &= \frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}} x}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e}+ f(\phi ) \\ \end{align*}

Where \(f(\phi )\) is used for the constant of integration since \(\phi \) is a function of both \(x\) and \(\phi \). Taking derivative of equation (3) w.r.t \(\phi \) gives

\begin{align*} \tag{4} \frac {\partial \phi }{\partial \phi } &= \text {Expression too large to display}+f'(\phi ) \\ &=0+f'(\phi ) \\ \end{align*}

But equation (2) says that \(\frac {\partial \phi }{\partial \phi } = \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}\). Therefore equation (4) becomes

\begin{equation} \tag{5} \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}} = 0+f'(\phi ) \end{equation}

Solving equation (5) for \( f'(\phi )\) gives

\[ f'(\phi ) = \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}} \]

Integrating the above w.r.t \(\phi \) gives

\begin{align*} \int f'(\phi ) \mathop {\mathrm {d}\phi } &= \int \left ( \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}\right ) \mathop {\mathrm {d}\phi } \\ f(\phi ) &= \int \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \phi + c_6 \\ \end{align*}

\[ \phi = \frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}} x}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e}+\int \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \phi + c_6 \]

But since \(\phi \) itself is a constant function, then let \(\phi =c_7\) where \(c_2\) is new constant and combining \(c_6\) and \(c_7\) constants into the constant \(c_6\) gives the solution as

\[ c_6 = \frac {\sqrt {2}\, \sqrt {8}\, \sqrt {\left (\frac {c_1 e \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}}{8}+n c \pi \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right )\right ) \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, e}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}} x}{\sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, {\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e}+\int \frac {{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\phi -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\phi -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \phi \]

Simplifying the above gives

\begin{align*} \frac {\int _{}^{\phi }\frac {{\left (4 \sqrt {\frac {2 e \left (\textit {\_a} -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {\frac {2 e \left (\textit {\_a} -V_{0} \right )+v_{0}^{2} m}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-32 n^{2} c^{2} m \left (e \left (\textit {\_a} -V_{0} \right )+\frac {v_{0}^{2} m}{2}\right ) \pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \textit {\_a} \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, {\left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-32 c^{2} \phi e m \,n^{2} \pi ^{2}+32 n^{2} c^{2} \left (-\frac {v_{0}^{2} m}{2}+e V_{0} \right ) m \,\pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e +\sqrt {2}\, \sqrt {8}\, \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 )}\, {\left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -c_1 e \right )}^{{1}/{4}} x}{\sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, {\left (4 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-32 c^{2} \phi e m \,n^{2} \pi ^{2}+32 n^{2} c^{2} \left (-\frac {v_{0}^{2} m}{2}+e V_{0} \right ) m \,\pi ^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e} &= c_6 \\ \end{align*}

Solving for \(\phi \) from the above solution(s) gives (after possible removing of solutions that do not verify)

\begin{align*} \phi &=\operatorname {RootOf}\left (-\sqrt {-\frac {2 e \left (-4 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, c \,m^{2} n \,v_{0}^{2}+8 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, V_{0} c e m n -8 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, \textit {\_Z} c e m n -c_1 e m \,v_{0}^{2}+2 V_{0} c_1 \,e^{2}-2 c_1 \textit {\_Z} \,e^{2}\right )}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m -c_1 e \right )}^{{1}/{4}} x -\int _{}^{\textit {\_Z}}\frac {{\left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \textit {\_a}}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \textit {\_a}}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_a} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \textit {\_a} \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_Z} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e +c_6 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_Z} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e \right )\\ \phi &=\operatorname {RootOf}\left (\sqrt {-\frac {2 e \left (-4 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, c \,m^{2} n \,v_{0}^{2}+8 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, V_{0} c e m n -8 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, \textit {\_Z} c e m n -c_1 e m \,v_{0}^{2}+2 V_{0} c_1 \,e^{2}-2 c_1 \textit {\_Z} \,e^{2}\right )}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m -c_1 e \right )}^{{1}/{4}} x -\int _{}^{\textit {\_Z}}\frac {{\left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \textit {\_a}}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \textit {\_a}}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_a} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \textit {\_a} \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_Z} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e +c_3 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_Z} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e \right ) \end{align*}

Summary of solutions found

\begin{align*} \phi &= \operatorname {RootOf}\left (-\sqrt {-\frac {2 e \left (-4 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, c \,m^{2} n \,v_{0}^{2}+8 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, V_{0} c e m n -8 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, \textit {\_Z} c e m n -c_1 e m \,v_{0}^{2}+2 V_{0} c_1 \,e^{2}-2 c_1 \textit {\_Z} \,e^{2}\right )}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m -c_1 e \right )}^{{1}/{4}} x -\int _{}^{\textit {\_Z}}\frac {{\left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \textit {\_a}}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \textit {\_a}}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_a} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \textit {\_a} \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_Z} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e +c_6 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_Z} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e \right ) \\ \phi &= \operatorname {RootOf}\left (\sqrt {-\frac {2 e \left (-4 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, c \,m^{2} n \,v_{0}^{2}+8 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, V_{0} c e m n -8 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, \textit {\_Z} c e m n -c_1 e m \,v_{0}^{2}+2 V_{0} c_1 \,e^{2}-2 c_1 \textit {\_Z} \,e^{2}\right )}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m -c_1 e \right )}^{{1}/{4}} x -\int _{}^{\textit {\_Z}}\frac {{\left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \textit {\_a}}{m}}\, \pi n c m -c_1 e \right )}^{{1}/{4}}}{{\left (4 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \textit {\_a}}{m}}\, \pi n c m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_a} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}}}d \textit {\_a} \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_Z} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e +c_3 \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, {\left (4 \pi n c \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 \textit {\_Z} e}{m}}\, m +c_1 e \right )}^{{1}/{4}} \left (-16 \pi ^{2} c^{2} m^{2} n^{2} v_{0}^{2}+32 \pi ^{2} V_{0} c^{2} e m \,n^{2}-32 \pi ^{2} \textit {\_Z} \,c^{2} e m \,n^{2}+c_1^{2} e^{2}\right )^{{1}/{4}} e \right ) \\ \end{align*}

2.6.4.2 second order ode can be made integrable

42.400 (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 \,\pi ^{2} n^{2} c^{2}}&= x +c_4 \end{align*}

Simplifying the above gives

\begin{align*} \frac {\sqrt {2}\, \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 )}{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}}\, \pi ^{2} c^{2} m \,n^{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 \,\pi ^{2} n^{2} c^{2}}&= x +c_5 \end{align*}

Simplifying the above gives

\begin{align*} -\frac {\sqrt {2}\, \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 )}{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}}\, \pi ^{2} c^{2} m \,n^{2}} &= x +c_5 \\ \end{align*}

Summary of solutions found

\begin{align*} \frac {\sqrt {2}\, \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 )}{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}}\, \pi ^{2} c^{2} m \,n^{2}} &= x +c_4 \\ \end{align*}

2.6.4.3 ✓ Maple. Time used: 0.059 (sec). Leaf size: 204

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 {2 \left (\textit {\_a} -V_{0} \right ) e +v_{0}^{2} m}{m}}}{4 \sqrt {\sqrt {\frac {2 \left (\textit {\_a} -V_{0} \right ) e +v_{0}^{2} m}{m}}\, e \left (\frac {c_1 \sqrt {2 e \left (-\textit {\_a} +V_{0} \right )-v_{0}^{2} m}}{16}+n c \left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right ) \pi \right )}}d \textit {\_a} -x -c_2 &= 0 \\ -e \int _{}^{\phi }\frac {\sqrt {\frac {2 \left (\textit {\_a} -V_{0} \right ) e +v_{0}^{2} m}{m}}}{4 \sqrt {\sqrt {\frac {2 \left (\textit {\_a} -V_{0} \right ) e +v_{0}^{2} m}{m}}\, e \left (\frac {c_1 \sqrt {2 e \left (-\textit {\_a} +V_{0} \right )-v_{0}^{2} m}}{16}+n c \left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right ) \pi \right )}}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) = 1/2*y(x)/(2*(x-V__0)*e+ 
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