12.2.11 problem 11

Internal problem ID [1547]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Linear first order. Section 2.1 Page 41
Problem number : 11
Date solved : Saturday, March 29, 2025 at 10:58:29 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.057 (sec). Leaf size: 18

ode:=diff(y(x),x)+tan(k*x)*y(x) = 0; 
ic:=y(0) = 2; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = 2 \left (\sec \left (k x \right )^{2}\right )^{-\frac {1}{2 k}} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 15

ode=D[y[x],x] +Tan[k*x]*y[x]==0; 
ic=y[0]==2; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to 2 \sqrt [k]{\cos (k x)} \]

✓ Sympy. Time used: 0.648 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = \begin {cases} 2 e^{\frac {\log {\left (\cos {\left (k x \right )} \right )}}{k}} & \text {for}\: k > 0 \vee k < 0 \\2 & \text {otherwise} \end {cases} \]