89.7.20 problem 20

Internal problem ID [24451]
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 : 20
Date solved : Sunday, October 12, 2025 at 05:55:28 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y \left (x \tan \left (x \right )+\ln \left (y\right )\right )+\tan \left (x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.297 (sec). Leaf size: 15

ode:=y(x)*(x*tan(x)+ln(y(x)))+tan(x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-1+\cot \left (x \right ) x +c_1 \csc \left (x \right )} \]

✓ Mathematica. Time used: 0.502 (sec). Leaf size: 21

ode=y[x]*( x*Tan[x]+Log[y[x]] )+Tan[x]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{x \cot (x)+\frac {1}{2} c_1 \csc (x)-1} \end{align*}

✓ Sympy. Time used: 1.729 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x*tan(x) + log(y(x)))*y(x) + tan(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = e^{\frac {C_{1}}{\sin {\left (x \right )}} + \frac {x}{\tan {\left (x \right )}} - 1} \]