Internal problem ID [12029]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page
368
Problem number: Problem 14(a).
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y=12 \operatorname {Heaviside}\left (t \right )-12 \operatorname {Heaviside}\left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 91
dsolve([diff(y(t),t$4)-5*diff(y(t),t$2)+4*y(t)=12*(Heaviside(t)-Heaviside(t-1)),y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = 2 \,{\mathrm e}^{-2 t} \left ({\mathrm e}^{3 t -1} \operatorname {Heaviside}\left (t -1\right )-\frac {{\mathrm e}^{4 t -2} \operatorname {Heaviside}\left (t -1\right )}{4}+\left (-\frac {{\mathrm e}^{2}}{4}-\frac {3 \,{\mathrm e}^{2 t}}{2}+{\mathrm e}^{1+t}\right ) \operatorname {Heaviside}\left (t -1\right )-\left ({\mathrm e}^{t}-\frac {3 \,{\mathrm e}^{2 t}}{2}+{\mathrm e}^{3 t}-\frac {{\mathrm e}^{4 t}}{4}-\frac {1}{4}\right ) \operatorname {Heaviside}\left (t \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 88
DSolve[{y''''[t]-5*y''[t]+4*y[t]==12*(UnitStep[t]-UnitStep[t-1]),{y[0]==0,y'[0]==0,y''[0]==0,y'''[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} e^{-2 t} \left (-1+e^t\right )^4 & 0\leq t\leq 1 \\ \frac {1}{2} (-1+e) e^{-2 (t+1)} \left (-e^2-e^3+e^{4 t}+4 e^{t+2}-4 e^{3 t+1}+e^{4 t+1}\right ) & t>1 \\ \end {array} \\ \end {array} \]