56.3.12 problem 12

Internal problem ID [8870]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 12
Date solved : Monday, January 27, 2025 at 05:09:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y^{\prime }\left (1\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.240 (sec). Leaf size: 110

dsolve([diff(y(x),x$2)+diff(y(x),x)+y(x)=sin(x),D(y)(1) = 0],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.343 (sec). Leaf size: 4176

DSolve[{D[y[x],{x,3}]+D[y[x],x]+y[x]==Sin[x],{Derivative[1][y][1] == 0}},y[x],x,IncludeSingularSolutions -> True]
 

Too large to display