80.6.4 problem 4 (eq 50)

Internal problem ID [18568]
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, January 28, 2025 at 12:00:26 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*}

✓ Solution by Maple

Time used: 0.073 (sec). Leaf size: 210

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

\begin{align*} e \left (\int _{}^{\phi \left (x \right )}\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}+\left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right ) \pi c n \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 \left (x \right )}\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}+\left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right ) \pi c n \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*}

✓ Solution by Mathematica

Time used: 79.952 (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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