23.4.53 problem 53

Internal problem ID [6355]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 53
Date solved : Tuesday, September 30, 2025 at 02:52:24 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y^{\prime }+4 {y^{\prime }}^{3}+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 49

ode:=2*diff(y(x),x)+4*diff(y(x),x)^3+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2 \,{\mathrm e}^{4 x} c_1 -4}}{2}\right )}{4}+c_2 \\ y &= -\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2 \,{\mathrm e}^{4 x} c_1 -4}}{2}\right )}{4}+c_2 \\ \end{align*}

✓ Mathematica. Time used: 60.073 (sec). Leaf size: 95

ode=2*D[y[x],x] + 4*D[y[x],x]^3 + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2-\frac {\arctan \left (\frac {e^{-c_1} \sqrt {e^{4 x}-2 e^{2 c_1}}}{\sqrt {2}}\right )}{2 \sqrt {2}}\\ y(x)&\to \frac {\arctan \left (\frac {e^{-c_1} \sqrt {e^{4 x}-2 e^{2 c_1}}}{\sqrt {2}}\right )}{2 \sqrt {2}}+c_2 \end{align*}

✓ Sympy. Time used: 24.220 (sec). Leaf size: 122

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*Derivative(y(x), x)**3 + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} - \int \sqrt {- \frac {C_{2}}{2 C_{2} - e^{4 x}}}\, dx, \ y{\left (x \right )} = C_{1} + \int \sqrt {- \frac {C_{2}}{2 C_{2} - e^{4 x}}}\, dx, \ y{\left (x \right )} = C_{1} - \int \sqrt {- \frac {C_{2}}{2 C_{2} - e^{4 x}}}\, dx, \ y{\left (x \right )} = C_{1} + \int \sqrt {- \frac {C_{2}}{2 C_{2} - e^{4 x}}}\, dx, \ y{\left (x \right )} = C_{1} - \int \sqrt {- \frac {C_{2}}{2 C_{2} - e^{4 x}}}\, dx, \ y{\left (x \right )} = C_{1} + \int \sqrt {- \frac {C_{2}}{2 C_{2} - e^{4 x}}}\, dx\right ] \]