35.3.5 problem 5

Internal problem ID [6109]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 3. Linear First-Order Equations. page 403
Problem number : 5
Date solved : Wednesday, March 05, 2025 at 12:12:52 AM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

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

\[ y = \frac {\left (x -\cos \left (x \right )+c_1 \right ) \left (\cos \left (x \right )-\sin \left (x \right )+1\right )}{\cos \left (x \right )+\sin \left (x \right )+1} \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 25

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

\[ y(x)\to e^{-2 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )} (x-\cos (x)+c_1) \]

✓ Sympy. Time used: 25.894 (sec). Leaf size: 138

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

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