82.44.4 problem Ex. 4

Internal problem ID [18889]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 99
Problem number : Ex. 4
Date solved : Thursday, March 13, 2025 at 01:08:15 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 49

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

\begin{align*} y \left (x \right ) &= \frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2 c_{1} {\mathrm e}^{4 x}-4}}{2}\right )}{4}+c_{2} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2 c_{1} {\mathrm e}^{4 x}-4}}{2}\right )}{4}+c_{2} \\ \end{align*}

✓ Mathematica. Time used: 60.108 (sec). Leaf size: 95

ode=D[y[x],{x,2}]+2*D[y[x],x]+4*D[y[x],x]^3==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: 38.263 (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)
 

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