76.1.10 problem 10

Internal problem ID [17309]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.1 (Separable equations). Problems at page 44
Problem number : 10
Date solved : Tuesday, January 28, 2025 at 09:59:23 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {\sec \left (x \right )^{2}}{y^{3}+1} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 17

dsolve(diff(y(x),x)=sec(x)^2/(1+y(x)^3),y(x), singsol=all)
 

\[ \tan \left (x \right )-\frac {y^{4}}{4}-y+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.278 (sec). Leaf size: 1075

DSolve[D[y[x],x]==Sec[x]^2/(1+y[x]^3),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt {\frac {-4\ 3^{2/3} \tan (x)+\sqrt [3]{3} \left (9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {\frac {8 (\tan (x)+c_1)}{\sqrt [3]{3} \sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}-\frac {2 \sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}{3^{2/3}}-\frac {4 \sqrt {6}}{\sqrt {\frac {-4\ 3^{2/3} \tan (x)+\sqrt [3]{3} \left (9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}}}} \\ y(x)\to \frac {\sqrt {\frac {-4\ 3^{2/3} \tan (x)+\sqrt [3]{3} \left (9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}}}{\sqrt {6}}+\frac {1}{2} \sqrt {\frac {8 (\tan (x)+c_1)}{\sqrt [3]{3} \sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}-\frac {2 \sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}{3^{2/3}}-\frac {4 \sqrt {6}}{\sqrt {\frac {-4\ 3^{2/3} \tan (x)+\sqrt [3]{3} \left (9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}}}} \\ y(x)\to -\frac {\sqrt {\frac {-4\ 3^{2/3} \tan (x)+\sqrt [3]{3} \left (9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {\frac {8 (\tan (x)+c_1)}{\sqrt [3]{3} \sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}-\frac {2 \sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}{3^{2/3}}+\frac {4 \sqrt {6}}{\sqrt {\frac {-4\ 3^{2/3} \tan (x)+\sqrt [3]{3} \left (9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}}}} \\ y(x)\to \frac {1}{2} \sqrt {\frac {8 (\tan (x)+c_1)}{\sqrt [3]{3} \sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}-\frac {2 \sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}{3^{2/3}}+\frac {4 \sqrt {6}}{\sqrt {\frac {-4\ 3^{2/3} \tan (x)+\sqrt [3]{3} \left (9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}}}}-\frac {\sqrt {\frac {-4\ 3^{2/3} \tan (x)+\sqrt [3]{3} \left (9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}\right ){}^{2/3}-4\ 3^{2/3} c_1}{\sqrt [3]{9+\sqrt {3} \sqrt {27+64 (\tan (x)+c_1){}^3}}}}}{\sqrt {6}} \\ \end{align*}