Internal problem ID [13476]
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).
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y=\tan \left (x \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 72
dsolve(diff(y(x),x$3)-diff(y(x),x$2)+diff(y(x),x)-y(x)=tan(x),y(x), singsol=all)
\[ y = \frac {\cos \left (x \right )^{2}}{2}+\frac {\cos \left (x \right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{2}-\left (\int -\frac {\left (\cos \left (x \right )^{2}+\sin \left (x \right )^{2}\right ) \tan \left (x \right ) {\mathrm e}^{-x}}{2}d x \right ) {\mathrm e}^{x}+\frac {\sin \left (x \right )^{2}}{2}-\frac {\sin \left (x \right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{2}+c_{1} \cos \left (x \right )+c_{2} {\mathrm e}^{x}+c_{3} \sin \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.06 (sec). Leaf size: 106
DSolve[y'''[x]-y''[x]+y'[x]-y[x]==Tan[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {1}{2} \sin (x) \text {arctanh}(\sin (x))+\frac {1}{2} \cos (x) \text {arctanh}(\sin (x))-\frac {1}{2} i \operatorname {Hypergeometric2F1}\left (\frac {i}{2},1,1+\frac {i}{2},-e^{2 i x}\right )-\left (\frac {1}{5}-\frac {i}{10}\right ) e^{2 i x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {i}{2},2+\frac {i}{2},-e^{2 i x}\right )+c_3 e^x+c_1 \cos (x)+c_2 \sin (x)+\frac {1}{2} \]