38.5.32 problem 32

Internal problem ID [8380]
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 : 32
Date solved : Tuesday, September 30, 2025 at 05:33:11 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (-2\right )&={\frac {1}{3}} \\ \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 42

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

\[ y = -\frac {2}{\sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, x}{\sqrt {\pi }}\right )+\sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}}{\sqrt {\pi }}\right )-6} \]

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 23

ode=D[y[x],x]==y[x]^2*Sin[x^2]; 
ic={y[-2]==1/3}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3-\int _{-2}^x\sin \left (K[1]^2\right )dK[1]} \end{align*}

✓ Sympy. Time used: 0.286 (sec). Leaf size: 54

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

\[ y{\left (x \right )} = - \frac {2}{\sqrt {2} \sqrt {\pi } S\left (\frac {\sqrt {2} x}{\sqrt {\pi }}\right ) - 6 + \sqrt {2} \sqrt {\pi } S\left (\frac {2 \sqrt {2}}{\sqrt {\pi }}\right )} \]