56.3.13 problem 13

Internal problem ID [8871]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 13
Date solved : Monday, January 27, 2025 at 05:10:08 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\\ y \left (2\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.075 (sec). Leaf size: 145

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

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

✓ Solution by Mathematica

Time used: 0.613 (sec). Leaf size: 12765

DSolve[{D[y[x],{x,3}]+D[y[x],x]+y[x]==Sin[x],{Derivative[1][y][1] == 0,y[2]==0}},y[x],x,IncludeSingularSolutions -> True]
 

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