7.3.13 problem 13

Internal problem ID [53]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.4 (separable equations). Problems at page 43
Problem number : 13
Date solved : Tuesday, March 04, 2025 at 10:40:46 AM
CAS classification : [_separable]

\begin{align*} y^{3} y^{\prime }&=\left (1+y^{4}\right ) \cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 63

ode:=y(x)^3*diff(y(x),x) = (1+y(x)^4)*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \left (c_1 \,{\mathrm e}^{4 \sin \left (x \right )}-1\right )^{{1}/{4}} \\ y &= -\left (c_1 \,{\mathrm e}^{4 \sin \left (x \right )}-1\right )^{{1}/{4}} \\ y &= -i \left (c_1 \,{\mathrm e}^{4 \sin \left (x \right )}-1\right )^{{1}/{4}} \\ y &= i \left (c_1 \,{\mathrm e}^{4 \sin \left (x \right )}-1\right )^{{1}/{4}} \\ \end{align*}

✓ Mathematica. Time used: 3.951 (sec). Leaf size: 153

ode=y[x]^3*D[y[x],x]==(1+y[x]^4)*Cos[x]; 
DSolve[ode,y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\sqrt [4]{-1+e^{4 (\sin (x)+c_1)}} \\ y(x)\to -i \sqrt [4]{-1+e^{4 (\sin (x)+c_1)}} \\ y(x)\to i \sqrt [4]{-1+e^{4 (\sin (x)+c_1)}} \\ y(x)\to \sqrt [4]{-1+e^{4 (\sin (x)+c_1)}} \\ y(x)\to -\sqrt [4]{-1} \\ y(x)\to \sqrt [4]{-1} \\ y(x)\to -(-1)^{3/4} \\ y(x)\to (-1)^{3/4} \\ y(x)\to -\frac {1+i}{\sqrt {2}} \\ y(x)\to \frac {1-i}{\sqrt {2}} \\ \end{align*}

✓ Sympy. Time used: 5.725 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-y(x)**4 - 1)*cos(x) + y(x)**3*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \left (-1\right )^{\frac {3}{4}} \sqrt [4]{C_{1} e^{4 \sin {\left (x \right )}} + 1}, \ y{\left (x \right )} = \left (-1\right )^{\frac {3}{4}} \sqrt [4]{C_{1} e^{4 \sin {\left (x \right )}} + 1}, \ y{\left (x \right )} = - \sqrt [4]{C_{1} e^{4 \sin {\left (x \right )}} - 1}, \ y{\left (x \right )} = \sqrt [4]{C_{1} e^{4 \sin {\left (x \right )}} - 1}\right ] \]