38.5.60 problem 50

Internal problem ID [8408]
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 : 50
Date solved : Tuesday, September 30, 2025 at 05:34:58 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 26

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

\[ y^{{3}/{2}}-3 \sin \left (\sqrt {x}\right )+3 \sqrt {x}\, \cos \left (\sqrt {x}\right )-c_1 = 0 \]

✓ Mathematica. Time used: 0.216 (sec). Leaf size: 34

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

\begin{align*} y(x)&\to \left (\frac {3}{2}\right )^{2/3} \left (\int _1^x\sin \left (\sqrt {K[1]}\right )dK[1]+c_1\right ){}^{2/3} \end{align*}

✓ Sympy. Time used: 17.781 (sec). Leaf size: 109

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

\[ \left [ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \left (C_{1} - 3 \sqrt {x} \cos {\left (\sqrt {x} \right )} + 3 \sin {\left (\sqrt {x} \right )}\right )^{\frac {2}{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \left (C_{1} - 3 \sqrt {x} \cos {\left (\sqrt {x} \right )} + 3 \sin {\left (\sqrt {x} \right )}\right )^{\frac {2}{3}}}{2}, \ y{\left (x \right )} = \left (C_{1} - 3 \sqrt {x} \cos {\left (\sqrt {x} \right )} + 3 \sin {\left (\sqrt {x} \right )}\right )^{\frac {2}{3}}\right ] \]