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 : Sunday, March 30, 2025 at 11:40:39 AM
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*}

✓ Maple. Time used: 0.164 (sec). Leaf size: 23

ode:=diff(y(x),x) = (1+y(x)^2)*(1+cos(x^3))^(1/2); 
ic:=y(1) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

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

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

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

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

✓ Sympy. Time used: 2.083 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-y(x)**2 - 1)*sqrt(cos(x**3) + 1) + Derivative(y(x), x),0) 
ics = {y(1): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \tan {\left (\int \sqrt {\cos {\left (x^{3} \right )} + 1}\, dx - \int \limits ^{1} \sqrt {\cos {\left (x^{3} \right )} + 1}\, dx + \frac {\pi }{4} \right )} \]