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 : Friday, November 28, 2025 at 06:42:54 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

Solved as second order missing x ode

Time used: 67.441 (sec)

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*}

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 \(p(\phi )\).

Solve The 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*}

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

Simplifying the above gives

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

Solving for \(p\) gives

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

Solve Integrating 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 (-e^{2} c_1 \sqrt {4 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, c e m n +c_1 \,e^{2}}+\frac {{\left (4 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, c e m n +c_1 \,e^{2}\right )}^{{3}/{2}}}{3}\right )}{16 e^{2} m \,\pi ^{2} c^{2} n^{2}}&= x +c_2 \end{align*}

Simplifying the above gives

\begin{align*} \frac {\sqrt {2}\, \sqrt {4 \pi \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, c e m n +c_1 \,e^{2}}\, \left (2 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -c_1 e \right )}{24 e m \,\pi ^{2} c^{2} n^{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 \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*}

Solve Integrating 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 (-e^{2} c_1 \sqrt {4 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, c e m n +c_1 \,e^{2}}+\frac {{\left (4 \pi \sqrt {-\frac {-v_{0}^{2} m +2 e V_{0} -2 e \phi }{m}}\, c e m n +c_1 \,e^{2}\right )}^{{3}/{2}}}{3}\right )}{16 e^{2} m \,\pi ^{2} c^{2} n^{2}}&= x +c_3 \end{align*}

Simplifying the above gives

\begin{align*} -\frac {\sqrt {2}\, \sqrt {4 \pi \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, c e m n +c_1 \,e^{2}}\, \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 )}{12 e m \,\pi ^{2} c^{2} n^{2}} &= x +c_3 \\ \end{align*}

Summary of solutions found

\begin{align*} \frac {\sqrt {2}\, \sqrt {4 \pi \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, c e m n +c_1 \,e^{2}}\, \left (2 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -c_1 e \right )}{24 e m \,\pi ^{2} c^{2} n^{2}} &= x +c_2 \\ -\frac {\sqrt {2}\, \sqrt {4 \pi \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, c e m n +c_1 \,e^{2}}\, \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 )}{12 e m \,\pi ^{2} c^{2} n^{2}} &= x +c_3 \\ \end{align*}

Solved as second order can be made integrable

Time used: 166.433 (sec)

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*}

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 \(\phi \). Solve 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)

Solve Integrating 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 (-e^{2} c_1 \sqrt {4 \pi c m n \sqrt {\frac {v_{0}^{2} m -2 e V_{0} +2 e \phi }{m}}\, e +c_1 \,e^{2}}+\frac {{\left (4 \pi c m n \sqrt {\frac {v_{0}^{2} m -2 e V_{0} +2 e \phi }{m}}\, e +c_1 \,e^{2}\right )}^{{3}/{2}}}{3}\right )}{16 e^{2} m \,\pi ^{2} c^{2} n^{2}}&= x +c_4 \end{align*}

Simplifying the above gives

\begin{align*} \frac {\sqrt {2}\, \sqrt {4 \pi \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, c e m n +c_1 \,e^{2}}\, \left (2 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -c_1 e \right )}{24 e m \,\pi ^{2} c^{2} n^{2}} &= x +c_4 \\ \end{align*}

Solving Eq. (2)

Solve Integrating 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 (-e^{2} c_1 \sqrt {4 \pi c m n \sqrt {\frac {v_{0}^{2} m -2 e V_{0} +2 e \phi }{m}}\, e +c_1 \,e^{2}}+\frac {{\left (4 \pi c m n \sqrt {\frac {v_{0}^{2} m -2 e V_{0} +2 e \phi }{m}}\, e +c_1 \,e^{2}\right )}^{{3}/{2}}}{3}\right )}{16 e^{2} m \,\pi ^{2} c^{2} n^{2}}&= x +c_5 \end{align*}

Simplifying the above gives

\begin{align*} -\frac {\sqrt {2}\, \sqrt {4 \pi \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, c e m n +c_1 \,e^{2}}\, \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 )}{12 e m \,\pi ^{2} c^{2} n^{2}} &= x +c_5 \\ \end{align*}

Summary of solutions found

\begin{align*} \frac {\sqrt {2}\, \sqrt {4 \pi \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, c e m n +c_1 \,e^{2}}\, \left (2 \pi n c \sqrt {\frac {v_{0}^{2} m +2 e \phi -2 e V_{0}}{m}}\, m -c_1 e \right )}{24 e m \,\pi ^{2} c^{2} n^{2}} &= x +c_4 \\ \end{align*}

✓ 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
 

✓ 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

✗ 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