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 : Friday, February 07, 2025 at 07:47:04 AM
CAS classification : [_separable]

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

✓ Solution by Maple

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

dsolve(y(x)^3*diff(y(x),x)=(1+y(x)^4)*cos(x),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*}

✓ Solution by Mathematica

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

DSolve[y[x]^3*D[y[x],x]==(1+y[x]^4)*Cos[x],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*}