Internal problem ID [6449]
Book: Own collection of miscellaneous problems
Section: section 3.0
Problem number: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y^{\prime }+y-\sin \relax (x )=0} \end {gather*} With initial conditions \begin {align*} [y^{\prime }\relax (1) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.108 (sec). Leaf size: 80
dsolve([diff(y(x),x$2)+diff(y(x),x)+y(x)=sin(x),D(y)(1) = 0],y(x), singsol=all)
\[ y \relax (x ) = \frac {2 \sin \relax (1) {\mathrm e}^{-\frac {x}{2}+\frac {1}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_{2} {\mathrm e}^{-\frac {x}{2}} \left (\sqrt {3}\, \cos \left (\frac {\sqrt {3}}{2}\right )-\sin \left (\frac {\sqrt {3}}{2}\right )\right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\left (\sin \left (\frac {\sqrt {3}}{2}\right ) \sqrt {3}+\cos \left (\frac {\sqrt {3}}{2}\right )\right ) \left (c_{2} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )-\cos \relax (x )\right )}{\sin \left (\frac {\sqrt {3}}{2}\right ) \sqrt {3}+\cos \left (\frac {\sqrt {3}}{2}\right )} \]
✓ Solution by Mathematica
Time used: 0.451 (sec). Leaf size: 337
DSolve[{y'''[x]+y'[x]+y[x]==Sin[x],{y'[1] == 0}},y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{186} \text {Root}\left [\text {$\#$1}^6-3 \text {$\#$1}^5+7 \text {$\#$1}^4-9 \text {$\#$1}^3+7 \text {$\#$1}^2-3 \text {$\#$1}+1\&,1\right ] \left (186 \left (c_1 \exp \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}-1\&,3\right ]+(x-2) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ]\right )+c_2 \text {Root}\left [\text {$\#$1}^6+3 \text {$\#$1}^5+7 \text {$\#$1}^4+9 \text {$\#$1}^3+7 \text {$\#$1}^2+3 \text {$\#$1}+1\&,4\right ] \exp \left ((x-1) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ]+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ]\right )+c_2 \text {Root}\left [\text {$\#$1}^6-3 \text {$\#$1}^5+7 \text {$\#$1}^4-9 \text {$\#$1}^3+7 \text {$\#$1}^2-3 \text {$\#$1}+1\&,4\right ] \exp \left (x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,2\right ]\right )+c_1 \text {Root}\left [\text {$\#$1}^6-3 \text {$\#$1}^5+7 \text {$\#$1}^4-9 \text {$\#$1}^3+7 \text {$\#$1}^2-3 \text {$\#$1}+1\&,4\right ] \exp \left (x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,1\right ]\right )\right )+6 \cos (1) \text {Root}\left [\text {$\#$1}^3+31 \text {$\#$1}^2+29791\&,1\right ] \exp \left ((x-1) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}+1\&,3\right ]\right )+31 \sin (x) \left (3+\text {Root}\left [\text {$\#$1}^6+117 \text {$\#$1}^4+1539 \text {$\#$1}^2+22599\&,4\right ]\right )\right ) \\ \end{align*}