8.3.15 problem 16

Internal problem ID [691]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.4. Separable equations. Page 43
Problem number : 16
Date solved : Saturday, March 29, 2025 at 10:13:43 PM
CAS classification : [_separable]

\begin{align*} \left (x^{2}+1\right ) \tan \left (y\right ) y^{\prime }&=x \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 16

ode:=(x^2+1)*tan(y(x))*diff(y(x),x) = x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arccos \left (\frac {1}{\sqrt {x^{2}+1}\, c_1}\right ) \]

✓ Mathematica. Time used: 14.703 (sec). Leaf size: 63

ode=(x^2+1)*Tan[y[x]]*D[y[x],x] == x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\arccos \left (\frac {e^{-c_1}}{\sqrt {x^2+1}}\right ) \\ y(x)\to \arccos \left (\frac {e^{-c_1}}{\sqrt {x^2+1}}\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}

✓ Sympy. Time used: 0.481 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + (x**2 + 1)*tan(y(x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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