80.6.4 problem 4 (eq 50)

Internal problem ID [18497]
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, March 31, 2025 at 05:37:44 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\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*}

✓ Maple. Time used: 0.086 (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 {e \sqrt {\frac {\left (-2 V_{0} +2 \textit {\_a} \right ) e +v_{0}^{2} m}{m}}\, \left (\frac {c_1 \sqrt {\left (-2 \textit {\_a} +2 V_{0} \right ) e -v_{0}^{2} m}}{16}+n \pi \left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right ) c \right )}}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 {e \sqrt {\frac {\left (-2 V_{0} +2 \textit {\_a} \right ) e +v_{0}^{2} m}{m}}\, \left (\frac {c_1 \sqrt {\left (-2 \textit {\_a} +2 V_{0} \right ) e -v_{0}^{2} m}}{16}+n \pi \left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right ) c \right )}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 79.952 (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]
 

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✗ 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