89.7.11 problem 11

Internal problem ID [24442]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 61
Problem number : 11
Date solved : Thursday, October 02, 2025 at 10:33:01 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} 3 \sin \left (y\right )-5 x +2 x^{2} \cot \left (y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 18

ode:=3*sin(y(x))-5*x+2*x^2*cot(y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arcsin \left (\frac {x^{{5}/{2}}}{x^{{3}/{2}}-c_1}\right ) \]

✓ Mathematica. Time used: 0.469 (sec). Leaf size: 26

ode=(3*Sin[y[x]]-5*x )+(2*x^2*Cot[y[x]] )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \csc ^{-1}\left (\frac {1}{x}+\frac {15 c_1}{4 x^{5/2}}\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.857 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*cot(y(x))*Derivative(y(x), x) - 5*x + 3*sin(y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {x^{\frac {5}{2}}}{C_{1} + x^{\frac {3}{2}}} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {x^{\frac {5}{2}}}{C_{1} + x^{\frac {3}{2}}} \right )}\right ] \]