44.5.67 problem 57

Internal problem ID [7129]
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 : 57
Date solved : Sunday, March 30, 2025 at 11:48:04 AM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{2}} \end{align*}

✓ Maple. Time used: 0.412 (sec). Leaf size: 31

ode:=diff(y(x),x) = (1+y(x)^2)^(1/2)*sin(y(x))^2; 
ic:=y(0) = 1/2; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (2 \int _{\frac {1}{2}}^{\textit {\_Z}}\frac {1}{\sqrt {\textit {\_a}^{2}+1}\, \left (-1+\cos \left (2 \textit {\_a} \right )\right )}d \textit {\_a} +x \right ) \]

✗ Mathematica

ode=D[y[x],x]==Sqrt[1+y[x]^2]*Sin[y[x]]^2; 
ic={y[0]==1/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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✓ Sympy. Time used: 1.199 (sec). Leaf size: 37

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

\[ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {y^{2} + 1} \sin ^{2}{\left (y \right )}}\, dy = x + \int \limits ^{\frac {1}{2}} \frac {1}{\sqrt {y^{2} + 1} \sin ^{2}{\left (y \right )}}\, dy \]