12.6.19 problem 19

Internal problem ID [1698]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Exact equations. Section 2.5 Page 79
Problem number : 19
Date solved : Tuesday, March 04, 2025 at 01:35:13 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class A`]]

\begin{align*} -4 y \cos \left (x \right )+4 \sin \left (x \right ) \cos \left (x \right )+\sec \left (x \right )^{2}+\left (4 y-4 \sin \left (x \right )\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{4}\right )&=0 \end{align*}

✓ Maple. Time used: 1.013 (sec). Leaf size: 32

ode:=-4*cos(x)*y(x)+4*sin(x)*cos(x)+sec(x)^2+(4*y(x)-4*sin(x))*diff(y(x),x) = 0; 
ic:=y(1/4*Pi) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 6.517 (sec). Leaf size: 38

ode=(-4*y[x]*Cos[x]+4*Sin[x]*Cos[x]+Sec[x]^2)+(4*y[x]-4*Sin[x])*D[y[x],x]==0; 
ic=y[Pi/4]==0; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \sin (x)+\frac {1}{2} \sqrt {-\sec ^2(x)} \sqrt {\sin (2 x)-2 \cos (2 x)-2} \]

✓ Sympy. Time used: 11.216 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((4*y(x) - 4*sin(x))*Derivative(y(x), x) - 4*y(x)*cos(x) + 4*sin(x)*cos(x) + cos(x)**(-2),0) 
ics = {y(pi/4): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {\sqrt {4 - 2 \tan {\left (x \right )}}}{2} + \sin {\left (x \right )} \]