39.3.1 problem Problem 12.1

Internal problem ID [6527]
Book : Schaums Outline Differential Equations, 4th edition. Bronson and Costa. McGraw Hill 2014
Section : Chapter 12. VARIATION OF PARAMETERS. page 104
Problem number : Problem 12.1
Date solved : Wednesday, March 05, 2025 at 12:55:49 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 72

ode:=diff(diff(diff(y(x),x),x),x)+diff(y(x),x) = sec(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {i \left ({\mathrm e}^{i x}-{\mathrm e}^{-i x}\right ) \ln \left (\frac {{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}+1}\right )}{2}-\frac {i {\mathrm e}^{-i x}}{2}-2 i \arctan \left ({\mathrm e}^{i x}\right )+\frac {i {\mathrm e}^{i x}}{2}+\left (1+c_1 -\ln \left (2\right )\right ) \sin \left (x \right )+\left (-x -c_2 \right ) \cos \left (x \right )+c_3 \]

✓ Mathematica. Time used: 0.086 (sec). Leaf size: 57

ode=D[y[x],{x,3}]+D[y[x],x]==Sec[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -(x+c_2) \cos (x)-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )+\sin (x) (\log (\cos (x))+c_1)+c_3 \]

✓ Sympy. Time used: 0.331 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) + Derivative(y(x), (x, 3)) - 1/cos(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + \left (C_{2} - x\right ) \cos {\left (x \right )} + \left (C_{3} + \log {\left (\cos {\left (x \right )} \right )}\right ) \sin {\left (x \right )} - \frac {\log {\left (\sin {\left (x \right )} - 1 \right )}}{2} + \frac {\log {\left (\sin {\left (x \right )} + 1 \right )}}{2} \]