2.6.4 problem 4 (eq 50)

Internal problem ID [18244]
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 : Monday, December 23, 2024 at 09:18:54 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

Solve

\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*}

Solved as second order missing x ode

Time used: 4.773 (sec)

This 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 \(p(\phi )\).

The ode \(p^{\prime } = \frac {4 \pi n c}{p \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}}\) is separable as it can be written as

\begin{align*} p^{\prime }&= \frac {4 \pi n c}{p \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}}\\ &= 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 }\\ \frac {p^{2}}{2}&=\frac {4 \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}\, \pi n c m}{e}+c_{1} \end{align*}

Solving for \(p\) gives

\begin{align*} p &= \frac {\sqrt {2}\, \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 )}}{e} \\ p &= -\frac {\sqrt {2}\, \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 )}}{e} \\ \end{align*}

For solution (1) found earlier, since \(p=\phi ^{\prime }\) then we now have a new first order ode to solve which is

\begin{align*} \phi ^{\prime } = \frac {\sqrt {2}\, \sqrt {e \left (4 \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \pi n c m +c_{1} e \right )}}{e} \end{align*}

Integrating gives

\begin{align*} \int \frac {e \sqrt {2}}{2 \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 )}}d \phi &= dx\\ \frac {\sqrt {2}\, \sqrt {4 e c m n \pi \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}+e^{2} c_{1}}\, \left (2 \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}\, \pi n c m -c_{1} e \right )}{24 e \,n^{2} m \,c^{2} \pi ^{2}}&= x +c_{2} \end{align*}

For solution (2) found earlier, since \(p=\phi ^{\prime }\) then we now have a new first order ode to solve which is

\begin{align*} \phi ^{\prime } = -\frac {\sqrt {2}\, \sqrt {e \left (4 \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \pi n c m +c_{1} e \right )}}{e} \end{align*}

Integrating gives

\begin{align*} \int -\frac {e \sqrt {2}}{2 \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 )}}d \phi &= dx\\ -\frac {\sqrt {4 e c m n \pi \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}+e^{2} c_{1}}\, \left (\sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}\, \pi n c m -\frac {c_{1} e}{2}\right ) \sqrt {2}}{12 e \,n^{2} m \,c^{2} \pi ^{2}}&= x +c_3 \end{align*}

Will add steps showing solving for IC soon.

Summary of solutions found

\begin{align*} \frac {\sqrt {2}\, \sqrt {4 e c m n \pi \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}+e^{2} c_{1}}\, \left (2 \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}\, \pi n c m -c_{1} e \right )}{24 e \,n^{2} m \,c^{2} \pi ^{2}} &= x +c_{2} \\ -\frac {\sqrt {4 e c m n \pi \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}+e^{2} c_{1}}\, \left (\sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}\, \pi n c m -\frac {c_{1} e}{2}\right ) \sqrt {2}}{12 e \,n^{2} m \,c^{2} \pi ^{2}} &= x +c_3 \\ \end{align*}

Solved as second order can be made integrable

Time used: 5.401 (sec)

Multiplying 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 \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, \pi n c m +c_{1} e \right )}}{e} \\ \tag{2} \phi ^{\prime }&=-\frac {\sqrt {2}\, \sqrt {e \left (4 \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, \pi n c m +c_{1} e \right )}}{e} \\ \end{align*}

Now each of the above is solved separately.

Solving Eq. (1)

Integrating gives

\begin{align*} \int \frac {e \sqrt {2}}{2 \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 )}}d \phi &= dx\\ \frac {\sqrt {2}\, \sqrt {4 e c m n \pi \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}+e^{2} c_{1}}\, \left (2 \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}\, \pi n c m -c_{1} e \right )}{24 e \,n^{2} m \,c^{2} \pi ^{2}}&= x +c_{2} \end{align*}

Solving Eq. (2)

Integrating gives

\begin{align*} \int -\frac {e \sqrt {2}}{2 \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 )}}d \phi &= dx\\ -\frac {\sqrt {4 e c m n \pi \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}+e^{2} c_{1}}\, \left (\sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}\, \pi n c m -\frac {c_{1} e}{2}\right ) \sqrt {2}}{12 e \,n^{2} m \,c^{2} \pi ^{2}}&= x +c_3 \end{align*}

Will add steps showing solving for IC soon.

Summary of solutions found

\begin{align*} \frac {\sqrt {2}\, \sqrt {4 e c m n \pi \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}+e^{2} c_{1}}\, \left (2 \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}\, \pi n c m -c_{1} e \right )}{24 e \,n^{2} m \,c^{2} \pi ^{2}} &= x +c_{2} \\ -\frac {\sqrt {4 e c m n \pi \sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}+e^{2} c_{1}}\, \left (\sqrt {\frac {\left (-2 V_{0} +2 \phi \right ) e +v_{0}^{2} m}{m}}\, \pi n c m -\frac {c_{1} e}{2}\right ) \sqrt {2}}{12 e \,n^{2} m \,c^{2} \pi ^{2}} &= x +c_3 \\ \end{align*}

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \phi ^{\prime \prime }=\frac {4 \pi n c}{\sqrt {v_{0}^{2}+\frac {2 e \left (\phi -V_{0} \right )}{m}}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \phi ^{\prime \prime } \\ \bullet & {} & \textrm {Define new dependent variable}\hspace {3pt} u \\ {} & {} & u \left (x \right )=\phi ^{\prime } \\ \bullet & {} & \textrm {Compute}\hspace {3pt} \phi ^{\prime \prime } \\ {} & {} & u^{\prime }\left (x \right )=\phi ^{\prime \prime } \\ \bullet & {} & \textrm {Use chain rule on the lhs}\hspace {3pt} \\ {} & {} & \phi ^{\prime } \left (\frac {d}{d \phi }u \left (\phi \right )\right )=\phi ^{\prime \prime } \\ \bullet & {} & \textrm {Substitute in the definition of}\hspace {3pt} u \\ {} & {} & u \left (\phi \right ) \left (\frac {d}{d \phi }u \left (\phi \right )\right )=\phi ^{\prime \prime } \\ \bullet & {} & \textrm {Make substitutions}\hspace {3pt} \phi ^{\prime }=u \left (\phi \right ),\phi ^{\prime \prime }=u \left (\phi \right ) \left (\frac {d}{d \phi }u \left (\phi \right )\right )\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & u \left (\phi \right ) \left (\frac {d}{d \phi }u \left (\phi \right )\right )=\frac {4 \pi n c}{\sqrt {v_{0}^{2}+\frac {2 e \left (\phi -V_{0} \right )}{m}}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} \phi \\ {} & {} & \int u \left (\phi \right ) \left (\frac {d}{d \phi }u \left (\phi \right )\right )d \phi =\int \frac {4 \pi n c}{\sqrt {v_{0}^{2}+\frac {2 e \left (\phi -V_{0} \right )}{m}}}d \phi +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {u \left (\phi \right )^{2}}{2}=-\frac {4 \left (-v_{0}^{2} m +2 e V_{0} -2 e \phi \right ) \pi n c}{e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}}+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (\phi \right ) \\ {} & {} & \left \{u \left (\phi \right )=\frac {\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} 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}}}, u \left (\phi \right )=-\frac {\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} 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 \} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (\phi \right ) \\ {} & {} & u \left (\phi \right )=\frac {\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} 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}}} \\ \bullet & {} & \textrm {Revert to original variables with substitution}\hspace {3pt} u \left (\phi \right )=\phi ^{\prime },\phi =\phi \\ {} & {} & \phi ^{\prime }=\frac {\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} 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}}} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \phi ^{\prime }=\frac {\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} 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}}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\phi ^{\prime } \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}}{\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right )}}=\frac {1}{e} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {\phi ^{\prime } \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}}{\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right )}}d x =\int \frac {1}{e}d x +\mathit {C2} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right ) \sqrt {\frac {\left (-v_{0}^{2} m -2 e \phi +2 e V_{0} \right ) \left (4 \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \pi n c m +\mathit {C1} e \right )}{m}}\, \left (2 \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \pi n c m -\mathit {C1} e \right )}{12 \sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right )}\, e \sqrt {\sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right )}\, m \,\pi ^{2} n^{2} c^{2}}=\frac {x}{e}+\mathit {C2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \phi \\ {} & {} & \phi =\frac {-v_{0}^{2} m^{2} n^{2} c^{2}+2 e V_{0} n^{2} m \,c^{2}+{\left (\frac {{\left (e \left (576 \pi ^{4} \mathit {C2}^{2} c^{4} e^{2} m^{2} n^{4}+1152 \pi ^{4} \mathit {C2} \,c^{4} e \,m^{2} n^{4} x +576 \pi ^{4} c^{4} m^{2} n^{4} x^{2}+24 \sqrt {2}\, c^{2} m \,n^{2} \sqrt {288 \mathit {C2}^{4} \pi ^{4} c^{4} e^{4} m^{2} n^{4}+1152 \mathit {C2}^{3} \pi ^{4} c^{4} e^{3} m^{2} n^{4} x +1728 \mathit {C2}^{2} \pi ^{4} c^{4} e^{2} m^{2} n^{4} x^{2}+1152 \mathit {C2} \,\pi ^{4} c^{4} e \,m^{2} n^{4} x^{3}+288 \pi ^{4} c^{4} m^{2} n^{4} x^{4}-\mathit {C1}^{3} \mathit {C2}^{2} e^{4}-2 \mathit {C1}^{3} \mathit {C2} \,e^{3} x -\mathit {C1}^{3} e^{2} x^{2}}\, \pi ^{2}-\mathit {C1}^{3} e^{2}\right )\right )}^{{1}/{3}}}{4 \pi }+\frac {\mathit {C1}^{2} e^{2}}{4 \pi {\left (e \left (576 \pi ^{4} \mathit {C2}^{2} c^{4} e^{2} m^{2} n^{4}+1152 \pi ^{4} \mathit {C2} \,c^{4} e \,m^{2} n^{4} x +576 \pi ^{4} c^{4} m^{2} n^{4} x^{2}+24 \sqrt {2}\, c^{2} m \,n^{2} \sqrt {288 \mathit {C2}^{4} \pi ^{4} c^{4} e^{4} m^{2} n^{4}+1152 \mathit {C2}^{3} \pi ^{4} c^{4} e^{3} m^{2} n^{4} x +1728 \mathit {C2}^{2} \pi ^{4} c^{4} e^{2} m^{2} n^{4} x^{2}+1152 \mathit {C2} \,\pi ^{4} c^{4} e \,m^{2} n^{4} x^{3}+288 \pi ^{4} c^{4} m^{2} n^{4} x^{4}-\mathit {C1}^{3} \mathit {C2}^{2} e^{4}-2 \mathit {C1}^{3} \mathit {C2} \,e^{3} x -\mathit {C1}^{3} e^{2} x^{2}}\, \pi ^{2}-\mathit {C1}^{3} e^{2}\right )\right )}^{{1}/{3}}}+\frac {\mathit {C1} e}{4 \pi }\right )}^{2}}{2 c^{2} e m \,n^{2}} \\ \bullet & {} & \textrm {Solve 2nd ODE for}\hspace {3pt} u \left (\phi \right ) \\ {} & {} & u \left (\phi \right )=-\frac {\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} 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}}} \\ \bullet & {} & \textrm {Revert to original variables with substitution}\hspace {3pt} u \left (\phi \right )=\phi ^{\prime },\phi =\phi \\ {} & {} & \phi ^{\prime }=-\frac {\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} 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}}} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \phi ^{\prime }=-\frac {\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} 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}}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\phi ^{\prime } \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}}{\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right )}}=-\frac {1}{e} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {\phi ^{\prime } \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}}{\sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right )}}d x =\int -\frac {1}{e}d x +\mathit {C2} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right ) \sqrt {\frac {\left (-v_{0}^{2} m -2 e \phi +2 e V_{0} \right ) \left (4 \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \pi n c m +\mathit {C1} e \right )}{m}}\, \left (2 \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \pi n c m -\mathit {C1} e \right )}{12 \sqrt {-2 e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right )}\, e \sqrt {\sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\, \left (-4 \pi n c m \,v_{0}^{2}+8 V_{0} \pi n c e -8 \phi \pi n c e -\mathit {C1} e \sqrt {-\frac {-v_{0}^{2} m -2 e \phi +2 e V_{0}}{m}}\right )}\, m \,\pi ^{2} n^{2} c^{2}}=-\frac {x}{e}+\mathit {C2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \phi \\ {} & {} & \phi =\frac {-v_{0}^{2} m^{2} n^{2} c^{2}+2 e V_{0} n^{2} m \,c^{2}+{\left (\frac {{\left (e \left (576 \pi ^{4} \mathit {C2}^{2} c^{4} e^{2} m^{2} n^{4}-1152 \pi ^{4} \mathit {C2} \,c^{4} e \,m^{2} n^{4} x +576 \pi ^{4} c^{4} m^{2} n^{4} x^{2}+24 \sqrt {2}\, c^{2} m \,n^{2} \sqrt {288 \mathit {C2}^{4} \pi ^{4} c^{4} e^{4} m^{2} n^{4}-1152 \mathit {C2}^{3} \pi ^{4} c^{4} e^{3} m^{2} n^{4} x +1728 \mathit {C2}^{2} \pi ^{4} c^{4} e^{2} m^{2} n^{4} x^{2}-1152 \mathit {C2} \,\pi ^{4} c^{4} e \,m^{2} n^{4} x^{3}+288 \pi ^{4} c^{4} m^{2} n^{4} x^{4}-\mathit {C1}^{3} \mathit {C2}^{2} e^{4}+2 \mathit {C1}^{3} \mathit {C2} \,e^{3} x -\mathit {C1}^{3} e^{2} x^{2}}\, \pi ^{2}-\mathit {C1}^{3} e^{2}\right )\right )}^{{1}/{3}}}{4 \pi }+\frac {\mathit {C1}^{2} e^{2}}{4 \pi {\left (e \left (576 \pi ^{4} \mathit {C2}^{2} c^{4} e^{2} m^{2} n^{4}-1152 \pi ^{4} \mathit {C2} \,c^{4} e \,m^{2} n^{4} x +576 \pi ^{4} c^{4} m^{2} n^{4} x^{2}+24 \sqrt {2}\, c^{2} m \,n^{2} \sqrt {288 \mathit {C2}^{4} \pi ^{4} c^{4} e^{4} m^{2} n^{4}-1152 \mathit {C2}^{3} \pi ^{4} c^{4} e^{3} m^{2} n^{4} x +1728 \mathit {C2}^{2} \pi ^{4} c^{4} e^{2} m^{2} n^{4} x^{2}-1152 \mathit {C2} \,\pi ^{4} c^{4} e \,m^{2} n^{4} x^{3}+288 \pi ^{4} c^{4} m^{2} n^{4} x^{4}-\mathit {C1}^{3} \mathit {C2}^{2} e^{4}+2 \mathit {C1}^{3} \mathit {C2} \,e^{3} x -\mathit {C1}^{3} e^{2} x^{2}}\, \pi ^{2}-\mathit {C1}^{3} e^{2}\right )\right )}^{{1}/{3}}}+\frac {\mathit {C1} e}{4 \pi }\right )}^{2}}{2 e \,c^{2} m \,n^{2}} \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 
-> 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 = [[-( 
   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]
 

Maple dsolve solution

Solving time : 0.074 (sec)
Leaf size : 210

dsolve(diff(diff(phi(x),x),x) = 4*Pi*n*c/(v__0^2+2*e/m*(phi(x)-V__0))^(1/2), 
       phi(x),singsol=all)
 

\begin{align*} e \left (\int _{}^{\phi }\frac {\sqrt {\frac {\left (-2 V_{0} +2 \textit {\_a} \right ) e +v_{0}^{2} m}{m}}}{4 \sqrt {e \left (\frac {c_{1} \sqrt {\left (2 V_{0} -2 \textit {\_a} \right ) e -v_{0}^{2} m}}{16}+c n \left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right ) \pi \right ) \sqrt {\frac {\left (-2 V_{0} +2 \textit {\_a} \right ) e +v_{0}^{2} m}{m}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -e \left (\int _{}^{\phi }\frac {\sqrt {\frac {\left (-2 V_{0} +2 \textit {\_a} \right ) e +v_{0}^{2} m}{m}}}{4 \sqrt {e \left (\frac {c_{1} \sqrt {\left (2 V_{0} -2 \textit {\_a} \right ) e -v_{0}^{2} m}}{16}+c n \left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right ) \pi \right ) \sqrt {\frac {\left (-2 V_{0} +2 \textit {\_a} \right ) e +v_{0}^{2} m}{m}}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}

Mathematica DSolve solution

Solving time : 73.891 (sec)
Leaf size : 2754

DSolve[{D[phi[x],{x,2}]==4*Pi*n*c/Sqrt[v0^2+2*e/m*(phi[x]-V0)],{}}, 
       phi[x],x,IncludeSingularSolutions->True]
 

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