81.1.1 problem 1 (a)

Internal problem ID [18525]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter I. Introduction. Exercises at page 13
Problem number : 1 (a)
Date solved : Monday, March 31, 2025 at 05:41:15 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }&=c \left (1+{y^{\prime }}^{2}\right ) \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 24

ode:=diff(diff(y(x),x),x) = c*(1+diff(y(x),x)^2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\ln \left (c_1 \sin \left (c x \right )-c_2 \cos \left (c x \right )\right )}{c} \]

✓ Mathematica. Time used: 4.504 (sec). Leaf size: 21

ode=D[y[x],{x,2}]==c*(1+D[y[x],x]^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2-\frac {\log (\cos (c x+c_1))}{c} \]

✓ Sympy. Time used: 1.026 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-c*(Derivative(y(x), x)**2 + 1) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} + \begin {cases} \frac {\log {\left (\tan ^{2}{\left (C_{1} + c x \right )} + 1 \right )}}{2 c} & \text {for}\: c \neq 0 \\x \tan {\left (C_{1} \right )} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = C_{1} + \begin {cases} \frac {\log {\left (\tan ^{2}{\left (C_{1} + c x \right )} + 1 \right )}}{2 c} & \text {for}\: c \neq 0 \\x \tan {\left (C_{1} \right )} & \text {otherwise} \end {cases}\right ] \]