Internal problem ID [14573]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number: 68.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=f \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = a] \end {align*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 52
dsolve([diff(y(t),t$2)+diff(y(t),t)-2*y(t)=f(t),y(0) = 0, D(y)(0) = a],y(t), singsol=all)
\[ y \left (t \right ) = \frac {\left (-\left (\int _{0}^{t}f \left (\textit {\_z1} \right ) {\mathrm e}^{2 \textit {\_z1}}d \textit {\_z1} \right )+\left (\int _{0}^{t}f \left (\textit {\_z1} \right ) {\mathrm e}^{-\textit {\_z1}}d \textit {\_z1} \right ) {\mathrm e}^{3 t}+a \left ({\mathrm e}^{3 t}-1\right )\right ) {\mathrm e}^{-2 t}}{3} \]
✓ Solution by Mathematica
Time used: 0.078 (sec). Leaf size: 123
DSolve[{y''[t]+y'[t]-2*y[t]==f[t],{y[0]==0,y'[0]==a}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {1}{3} e^{-2 t} \left (3 \int _1^t-\frac {1}{3} e^{2 K[1]} f(K[1])dK[1]-3 e^{3 t} \int _1^0\frac {1}{3} e^{-K[2]} f(K[2])dK[2]+3 e^{3 t} \int _1^t\frac {1}{3} e^{-K[2]} f(K[2])dK[2]-3 \int _1^0-\frac {1}{3} e^{2 K[1]} f(K[1])dK[1]+a e^{3 t}-a\right ) \]