Internal problem ID [12498]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 5. The Laplace Transform Method. Exercises 5.5, page 273
Problem number: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+5 y=\cos \left (x \right )+2 \left (\delta \left (x -\pi \right )\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 50
dsolve([diff(y(x),x$2)-2*diff(y(x),x)+5*y(x)=cos(x)+2*Dirac(x-Pi),y(0) = 1, D(y)(0) = 0],y(x), singsol=all)
\[ y \left (x \right ) = \sin \left (2 x \right ) \operatorname {Heaviside}\left (x -\pi \right ) {\mathrm e}^{x -\pi }-\frac {7 \,{\mathrm e}^{x} \sin \left (2 x \right )}{20}+\frac {4 \,{\mathrm e}^{x} \cos \left (2 x \right )}{5}+\frac {\cos \left (x \right )}{5}-\frac {\sin \left (x \right )}{10} \]
✓ Solution by Mathematica
Time used: 0.506 (sec). Leaf size: 54
DSolve[{y''[x]-2*y'[x]+5*y[x]==Cos[x]+2*DiracDelta[x-Pi],{y[0]==1,y'[0]==0}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{10} \left (10 e^{x-\pi } \theta (x-\pi ) \sin (2 x)-\sin (x)+8 e^x \cos (2 x)+\left (2-7 e^x \sin (x)\right ) \cos (x)\right ) \]