68.4.44 problem 44

Internal problem ID [17224]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 44
Date solved : Thursday, October 02, 2025 at 01:58:49 PM
CAS classification : [_quadrature]

\begin{align*} \sin \left (y \right )^{2}&=x^{\prime } \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 14

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

\[ x = \frac {y}{2}-\frac {\sin \left (2 y \right )}{4} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 17

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

\begin{align*} x(y)&\to \int _0^y\sin ^2(K[1])dK[1] \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 14

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

\[ x{\left (y \right )} = \frac {y}{2} - \frac {\sin {\left (y \right )} \cos {\left (y \right )}}{2} \]