2.6.4.2 second order ode can be made integrable

\[\begin {aligned} \phi ^{\prime \prime }&=\frac {4 \pi n c}{\sqrt {v_{0}^{2}+\frac {2 e \left (\phi -V_{0} \right )}{m}}} \end {aligned}\]

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 \(\phi \).

Entering first order ode dAlembert solverLet \(p=\phi ^{\prime }\) the ode becomes

\begin{align*} \frac {p^{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*}

Solving for \(\phi \) from the above results in

\begin{align*} \phi &= \frac {-64 v_{0}^{2} m^{2} \pi ^{2} c^{2} n^{2}+128 e V_{0} \pi ^{2} c^{2} m \,n^{2}+e^{2} p^{4}-4 c_1 \,e^{2} p^{2}+4 c_1^{2} e^{2}}{128 \pi ^{2} c^{2} e m \,n^{2}}\tag {1A} \end{align*}

This has the form

\begin{align*} \phi &=x f(p)+g(p)\tag {*} \end{align*}

Where \(f,g\) are functions of \(p=\phi '(x)\). The above ode is dAlembert ode which is now solved.

Taking derivative of (*) w.r.t. \(x\) gives

\begin{align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end{align*}

Comparing the form \(\phi =x f + g\) to (1A) shows that

\begin{align*} f &= 0\\ g &= \frac {\left (p^{2}-2 c_1 \right )^{2} e^{2}+128 e V_{0} \pi ^{2} c^{2} m \,n^{2}-64 v_{0}^{2} m^{2} \pi ^{2} c^{2} n^{2}}{128 \pi ^{2} c^{2} e m \,n^{2}} \end{align*}

Hence (2) becomes

\begin{align*} p = \left (\frac {e \,p^{3}}{32 \pi ^{2} c^{2} m \,n^{2}}-\frac {e p c_1}{16 \pi ^{2} c^{2} m \,n^{2}}\right ) p^{\prime }\left (x \right )\tag {3} \end{align*}

The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives

\begin{align*} p = 0 \end{align*}

No valid singular solutions found.

The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\) from eq. (3). This results in

\begin{align*} p^{\prime }\left (x \right ) = \frac {p \left (x \right )}{\frac {e p \left (x \right )^{3}}{32 \pi ^{2} c^{2} m \,n^{2}}-\frac {e p \left (x \right ) c_1}{16 \pi ^{2} c^{2} m \,n^{2}}}\tag {4} \end{align*}

This ODE is now solved for \(p \left (x \right )\). No inversion is needed.

Integrating gives

\begin{align*} \int \frac {e \left (p^{2}-2 c_1 \right )}{32 c^{2} n^{2} m \,\pi ^{2}}d p &= dx\\ \frac {e \left (\frac {1}{3} p^{3}-2 c_1 p \right )}{32 c^{2} n^{2} m \,\pi ^{2}}&= x +c_2 \end{align*}

Solving for \(p\) from above gives

\[ p = \frac {{\left ({\left (\frac {{\left (\left (48 \pi ^{2} c_2 \,c^{2} m \,n^{2}+48 \pi ^{2} c^{2} m \,n^{2} x +2 \sqrt {2}\, \sqrt {288 \pi ^{4} c_2^{2} c^{4} m^{2} n^{4}+576 \pi ^{4} c_2 \,c^{4} m^{2} n^{4} x +288 \pi ^{4} c^{4} m^{2} n^{4} x^{2}-c_1^{3} e^{2}}\right ) e^{2}\right )}^{{1}/{3}}}{e}+\frac {2 c_1 e}{{\left (\left (48 \pi ^{2} c_2 \,c^{2} m \,n^{2}+48 \pi ^{2} c^{2} m \,n^{2} x +2 \sqrt {2}\, \sqrt {288 \pi ^{4} c_2^{2} c^{4} m^{2} n^{4}+576 \pi ^{4} c_2 \,c^{4} m^{2} n^{4} x +288 \pi ^{4} c^{4} m^{2} n^{4} x^{2}-c_1^{3} e^{2}}\right ) e^{2}\right )}^{{1}/{3}}}\right )}^{2}-2 c_1 \right )}^{2} e^{2}+128 e V_{0} \pi ^{2} c^{2} m \,n^{2}-64 v_{0}^{2} m^{2} \pi ^{2} c^{2} n^{2}}{128 \pi ^{2} c^{2} e m \,n^{2}} \]

Substituing the above solution for \(p\) in \(\phi = \frac {-64 v_{0}^{2} m^{2} \pi ^{2} c^{2} n^{2}+128 e V_{0} \pi ^{2} c^{2} m \,n^{2}+e^{2} p^{4}-4 c_1 \,e^{2} p^{2}+4 c_1^{2} e^{2}}{128 \pi ^{2} c^{2} e m \,n^{2}}\) gives

\[ \phi = \frac {{\left ({\left (\frac {{\left (\left (48 \pi ^{2} c_2 \,c^{2} m \,n^{2}+48 \pi ^{2} c^{2} m \,n^{2} x +2 \sqrt {2}\, \sqrt {288 \pi ^{4} c_2^{2} c^{4} m^{2} n^{4}+576 \pi ^{4} c_2 \,c^{4} m^{2} n^{4} x +288 \pi ^{4} c^{4} m^{2} n^{4} x^{2}-c_1^{3} e^{2}}\right ) e^{2}\right )}^{{1}/{3}}}{e}+\frac {2 c_1 e}{{\left (\left (48 \pi ^{2} c_2 \,c^{2} m \,n^{2}+48 \pi ^{2} c^{2} m \,n^{2} x +2 \sqrt {2}\, \sqrt {288 \pi ^{4} c_2^{2} c^{4} m^{2} n^{4}+576 \pi ^{4} c_2 \,c^{4} m^{2} n^{4} x +288 \pi ^{4} c^{4} m^{2} n^{4} x^{2}-c_1^{3} e^{2}}\right ) e^{2}\right )}^{{1}/{3}}}\right )}^{2}-2 c_1 \right )}^{2} e^{2}+128 e V_{0} \pi ^{2} c^{2} m \,n^{2}-64 v_{0}^{2} m^{2} \pi ^{2} c^{2} n^{2}}{128 \pi ^{2} c^{2} e m \,n^{2}} \]