68.4.25 problem 25

Internal problem ID [17205]
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 : 25
Date solved : Thursday, October 02, 2025 at 01:53:09 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

ode:=x*sin(x^2) = cos(y(x)^(1/2))/y(x)^(1/2)*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ -\frac {\cos \left (x^{2}\right )}{2}-2 \sin \left (\sqrt {y}\right )+c_1 = 0 \]

✓ Mathematica. Time used: 1.005 (sec). Leaf size: 63

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

\begin{align*} y(x)&\to \arcsin \left (\frac {1}{2} \left (-\int _1^xK[1] \sin \left (K[1]^2\right )dK[1]-c_1\right )\right ){}^2\\ y(x)&\to \arcsin \left (\frac {1}{2} \left (\int _1^xK[1] \sin \left (K[1]^2\right )dK[1]+c_1\right )\right ){}^2 \end{align*}

✓ Sympy. Time used: 1.490 (sec). Leaf size: 48

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

\[ \left [ y{\left (x \right )} = \operatorname {asin}^{2}{\left (\frac {C_{1}}{2} - \frac {\cos {\left (x^{2} \right )}}{4} \right )} - 2 \pi \operatorname {asin}{\left (\frac {C_{1}}{2} - \frac {\cos {\left (x^{2} \right )}}{4} \right )} + \pi ^{2}, \ y{\left (x \right )} = \operatorname {asin}^{2}{\left (C_{1} - \frac {\cos {\left (x^{2} \right )}}{4} \right )}\right ] \]