Internal problem ID [14627]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number: 60.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y=f \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 2] \end {align*}
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 47
dsolve([diff(y(t),t$2)+4*y(t)=f(t),y(0) = 0, D(y)(0) = 2],y(t), singsol=all)
\[ y \left (t \right ) = \sin \left (2 t \right )+\frac {\left (\int _{0}^{t}\cos \left (2 \textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \right ) \sin \left (2 t \right )}{2}-\frac {\left (\int _{0}^{t}\sin \left (2 \textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \right ) \cos \left (2 t \right )}{2} \]
✓ Solution by Mathematica
Time used: 0.089 (sec). Leaf size: 106
DSolve[{y''[t]+4*y[t]==f[t],{y[0]==0,y'[0]==2}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \sin (2 t) \left (-\int _1^0\frac {1}{2} \cos (2 K[2]) f(K[2])dK[2]\right )+\sin (2 t) \int _1^t\frac {1}{2} \cos (2 K[2]) f(K[2])dK[2]-\cos (2 t) \int _1^0-\cos (K[1]) f(K[1]) \sin (K[1])dK[1]+\cos (2 t) \int _1^t-\cos (K[1]) f(K[1]) \sin (K[1])dK[1]+\sin (2 t) \]