84.25.1 problem 15.1

Internal problem ID [22263]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 15. Variation of parameteres. Solved problems. Page 77
Problem number : 15.1
Date solved : Thursday, October 02, 2025 at 08:36:50 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 133

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

\[ y = -\frac {{\mathrm e}^{-x} \left (\left (\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}\, \int {\mathrm e}^{-\frac {x}{2}} \left (\sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{2}\right )+3 \sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \sec \left (x \right )d x -\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}\, \int -\left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{2}\right )-3 \cos \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \sec \left (x \right ) {\mathrm e}^{-\frac {x}{2}}d x -9 \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_2 -9 \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_3 \right ) {\mathrm e}^{\frac {3 x}{2}}-3 \int {\mathrm e}^{x} \sec \left (x \right )d x -9 c_1 \right )}{9} \]

✓ Mathematica. Time used: 0.355 (sec). Leaf size: 311

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

\begin{align*} y(x)&\to \frac {1}{6} \left ((2-2 i) e^{i x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},-e^{2 i x}\right )+2 \left ((1+2 i)+i \sqrt {3}\right ) e^{i x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} \left ((2+i)-\sqrt {3}\right ),\frac {1}{4} \left ((6+i)-\sqrt {3}\right ),-e^{2 i x}\right )-2 i \sqrt {3} e^{i x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} \left ((2+i)+\sqrt {3}\right ),\frac {1}{4} \left ((6+i)+\sqrt {3}\right ),-e^{2 i x}\right )+(2+4 i) e^{i x} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} \left ((2+i)+\sqrt {3}\right ),\frac {1}{4} \left ((6+i)+\sqrt {3}\right ),-e^{2 i x}\right )+6 c_1 e^{-x}+3 i c_2 e^{\frac {1}{2} \left (x-i \sqrt {3} x\right )}-3 i c_2 e^{\frac {1}{2} \left (x+i \sqrt {3} x\right )}+3 c_3 e^{\frac {1}{2} \left (x-i \sqrt {3} x\right )}+3 c_3 e^{\frac {1}{2} \left (x+i \sqrt {3} x\right )}\right ) \end{align*}

✓ Sympy. Time used: 17.678 (sec). Leaf size: 100

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

\[ y{\left (x \right )} = \left (C_{3} + \frac {\int e^{x} \sec {\left (x \right )}\, dx}{3}\right ) e^{- x} + \left (\left (C_{1} - \frac {2 \int e^{- \frac {x}{2}} \sin {\left (\frac {3 \sqrt {3} x + \pi }{6} \right )} \sec {\left (x \right )}\, dx}{3}\right ) \cos {\left (\frac {\sqrt {3} x}{2} \right )} + \left (C_{2} + \frac {2 \int e^{- \frac {x}{2}} \cos {\left (\frac {3 \sqrt {3} x + \pi }{6} \right )} \sec {\left (x \right )}\, dx}{3}\right ) \sin {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{\frac {x}{2}} \]