73.16.20 problem 24.4 (b)

Internal problem ID [15532]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 24. Variation of parameters. Additional exercises page 444
Problem number : 24.4 (b)
Date solved : Tuesday, January 28, 2025 at 08:01:23 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 46

dsolve(diff(y(x),x$3)-diff(y(x),x$2)+diff(y(x),x)-y(x)=tan(x),y(x), singsol=all)
 

\[ y = \frac {1}{2}+\frac {\left (\cos \left (x \right )-\sin \left (x \right )\right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{2}+{\mathrm e}^{x} c_{2} +\cos \left (x \right ) c_{1} +c_{3} \sin \left (x \right )+\frac {\left (\int \tan \left (x \right ) {\mathrm e}^{-x}d x \right ) {\mathrm e}^{x}}{2} \]

✓ Solution by Mathematica

Time used: 0.246 (sec). Leaf size: 97

DSolve[D[y[x],{x,3}]-D[y[x],{x,2}]+D[y[x],x]-y[x]==Tan[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^x \int _1^x\frac {1}{2} e^{-K[3]} \tan (K[3])dK[3]+\cos (x) \int _1^x\frac {1}{2} (\sin (K[1]) \tan (K[1])-\sin (K[1]))dK[1]+\sin (x) \int _1^x-\frac {1}{2} (\cos (K[2])+\sin (K[2])) \tan (K[2])dK[2]+c_3 e^x+c_1 \cos (x)+c_2 \sin (x) \]