89.7.14 problem 14

Internal problem ID [24445]
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 : 14
Date solved : Thursday, October 02, 2025 at 10:35:13 PM
CAS classification : [[_homogeneous, `class D`], _Bernoulli]

\begin{align*} x y^{\prime }&=y-y^{3} \cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 38

ode:=x*diff(y(x),x) = y(x)-y(x)^3*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x}{\sqrt {2 \cos \left (x \right )+2 x \sin \left (x \right )+c_1}} \\ y &= -\frac {x}{\sqrt {2 \cos \left (x \right )+2 x \sin \left (x \right )+c_1}} \\ \end{align*}

✓ Mathematica. Time used: 0.166 (sec). Leaf size: 51

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

\begin{align*} y(x)&\to -\frac {x}{\sqrt {2 x \sin (x)+2 \cos (x)+c_1}}\\ y(x)&\to \frac {x}{\sqrt {2 x \sin (x)+2 \cos (x)+c_1}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.911 (sec). Leaf size: 46

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

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