44.5.33 problem 33

Internal problem ID [7095]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 33
Date solved : Tuesday, January 28, 2025 at 08:49:00 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (1\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.352 (sec). Leaf size: 33

dsolve([diff(y(x),x)=(1+y(x)^2)*sqrt(1+cos(x^3)),y(1) = 1],y(x), singsol=all)
 

\[ y = \tan \left (\sqrt {2}\, \left (\int _{1}^{x}\operatorname {csgn}\left (\cos \left (\frac {\textit {\_z1}^{3}}{2}\right )\right ) \cos \left (\frac {\textit {\_z1}^{3}}{2}\right )d \textit {\_z1} \right )+\frac {\pi }{4}\right ) \]

✓ Solution by Mathematica

Time used: 30.553 (sec). Leaf size: 135

DSolve[{D[y[x],x]==(1+y[x]^2)*Sqrt[1+Cos[x^3]],{y[1]==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \tan \left (\frac {x^5 \left (3 \pi +2 \sqrt [6]{-1} 2^{5/6} \Gamma \left (\frac {1}{3},-\frac {i}{2}\right )-2 (-2)^{5/6} \Gamma \left (\frac {1}{3},\frac {i}{2}\right )\right )-2\ 2^{5/6} \sqrt [3]{i x^3} \left (x^6\right )^{2/3} \Gamma \left (\frac {1}{3},-\frac {i x^3}{2}\right )-2\ 2^{5/6} \sqrt [3]{-i x^3} \left (x^6\right )^{2/3} \Gamma \left (\frac {1}{3},\frac {i x^3}{2}\right )}{12 x^5}\right ) \]