75.16.73 problem 546

Internal problem ID [17046]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Trial and error method. Exercises page 132
Problem number : 546
Date solved : Tuesday, January 28, 2025 at 09:47:32 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 43

dsolve(diff(y(x),x$3)-y(x)=sin(x),y(x), singsol=all)
 

\[ y = c_{2} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_{3} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{x} c_{1} -\frac {\sin \left (x \right )}{2}+\frac {\cos \left (x \right )}{2} \]

✓ Solution by Mathematica

Time used: 0.235 (sec). Leaf size: 224

DSolve[D[y[x],{x,3}]-y[x]==Sin[x],y[x],x,IncludeSingularSolutions -> True]
 

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