81.9.1 problem 13-8

Internal problem ID [21590]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 13. The Wronskian and linear independence. Page 283.
Problem number : 13-8
Date solved : Thursday, October 02, 2025 at 07:58:46 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} u^{\prime \prime }+\left (\tan \left (x \right )-2 \cos \left (x \right )\right ) u^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 13

ode:=diff(diff(u(x),x),x)+(tan(x)-2*cos(x))*diff(u(x),x) = 0; 
dsolve(ode,u(x), singsol=all);
 

\[ u = c_1 +{\mathrm e}^{2 \sin \left (x \right )} c_2 \]

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 29

ode=D[u[x],{x,2}]+(Tan[x]-2*Cos[x])*D[u[x],x]==0; 
ic={}; 
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
 

\begin{align*} u(x)&\to \frac {1}{2} c_1 e^{-\text {arctanh}(\sin (x))+2 \sin (x)+\coth ^{-1}(\sin (x))}+c_2 \end{align*}

✓ Sympy. Time used: 0.203 (sec). Leaf size: 12

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

\[ u{\left (x \right )} = C_{1} + C_{2} e^{2 \sin {\left (x \right )}} \]