72.21.4 problem 4

Internal problem ID [14974]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 6. Laplace transform. Section 6.6. page 624
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 07:26:02 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+3 y&=\left (1-\operatorname {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}} \sin \left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 13.524 (sec). Leaf size: 190

dsolve([diff(y(t),t$2)+diff(y(t),t)+3*y(t)=(1-Heaviside(t-2))*exp(-(t-2)/10)*sin(t-2),y(0) = 1, D(y)(0) = 2],y(t), singsol=all)
 

\[ y = \frac {8000 \left (\left (\cos \left (t \right )-\frac {191 \sin \left (t \right )}{80}\right ) \cos \left (2\right )+\frac {191 \sin \left (2\right ) \left (\cos \left (t \right )+\frac {80 \sin \left (t \right )}{191}\right )}{80}\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}}}{42881}+\frac {100 \left (11 \left (191 \sin \left (2\right )+80 \cos \left (2\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )-318 \left (\cos \left (2\right )-\frac {782 \sin \left (2\right )}{795}\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \sqrt {11}\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{2}}}{471691}+\left (-\frac {4000}{42881}+\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (t -2\right )}+\left (-\frac {4000}{42881}-\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (t -2\right )}+\frac {200 \operatorname {Heaviside}\left (t -2\right ) \left (\left (-159 \sin \left (\sqrt {11}\right ) \sqrt {11}-440 \cos \left (\sqrt {11}\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (159 \sqrt {11}\, \cos \left (\sqrt {11}\right )-440 \sin \left (\sqrt {11}\right )\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{-\frac {t}{2}+1}}{471691}+\frac {5 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{11}+{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \]

✓ Solution by Mathematica

Time used: 13.046 (sec). Leaf size: 1335

DSolve[{D[y[t],{t,2}]+D[y[t],t]+8*y[t]==(1-UnitStep[t-2])*Exp[-(t-2)/10]*Sin[t-2],{y[0]==1,Derivative[1][y][0] ==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -\frac {e^{\frac {\sqrt {31} t-(2-135 i) t+(8-20 i) \sqrt {31}+930 i}{20-10 \sqrt {31}}} \left (4 e^{\frac {\left ((8+135 i)-4 \sqrt {31}\right ) t+(8-40 i) \sqrt {31}-(20-620 i)}{10 \left (-2+\sqrt {31}\right )}} \left (\left (100 \left (1240+659 i \sqrt {31}\right ) e^{\frac {(1+4 i) \sqrt {31}}{-2+\sqrt {31}}}-15000311 e^{\frac {(2+31 i)+2 i \sqrt {31}}{-2+\sqrt {31}}}+100 \left (1240-659 i \sqrt {31}\right ) e^{\frac {62 i+\sqrt {31}}{-2+\sqrt {31}}}-(124000+1071050 i) e^{\frac {(8+175 i)+\sqrt {31}}{5 \left (-2+\sqrt {31}\right )}}-(124000-1071050 i) e^{\frac {(8+135 i)+(1+20 i) \sqrt {31}}{5 \left (-2+\sqrt {31}\right )}}\right ) \cos \left (\frac {\sqrt {31} t}{2}\right )-5 \left (20 \left (-1240 i+659 \sqrt {31}\right ) e^{\frac {(1+4 i) \sqrt {31}}{-2+\sqrt {31}}}+483881 \sqrt {31} e^{\frac {(2+31 i)+2 i \sqrt {31}}{-2+\sqrt {31}}}+20 \left (1240 i+659 \sqrt {31}\right ) e^{\frac {62 i+\sqrt {31}}{-2+\sqrt {31}}}-(13180-7128 i) \sqrt {31} e^{\frac {(8+175 i)+\sqrt {31}}{5 \left (-2+\sqrt {31}\right )}}-(13180+7128 i) \sqrt {31} e^{\frac {(8+135 i)+(1+20 i) \sqrt {31}}{5 \left (-2+\sqrt {31}\right )}}\right ) \sin \left (\frac {\sqrt {31} t}{2}\right )\right )+10 e^{\frac {(2+10 i) \sqrt {31} t+20 i t-(5-20 i) \sqrt {31}+(10-310 i)}{10-5 \sqrt {31}}} \left (\cos \left (\frac {\sqrt {31} t}{2}\right )+i \sin \left (\frac {\sqrt {31} t}{2}\right )\right ) \left (\left (40 i \left (1240 i+659 \sqrt {31}\right ) e^{\frac {\left ((8+175 i)+20 i \sqrt {31}\right ) t+8 \sqrt {31}+620 i}{10 \left (-2+\sqrt {31}\right )}}+(80-691 i) \left (155+(10+4 i) \sqrt {31}\right ) e^{\frac {\left (175 i+2 \sqrt {31}\right ) t+20 i \sqrt {31}+(8+135 i)}{5 \left (-2+\sqrt {31}\right )}}-(80-691 i) \left (155+(10+4 i) \sqrt {31}\right ) e^{\frac {\left (20 i+(2+10 i) \sqrt {31}\right ) t+20 i \sqrt {31}+(8+135 i)}{5 \left (-2+\sqrt {31}\right )}}+(160+1382 i) \left (155-(10-4 i) \sqrt {31}\right ) e^{\frac {\left (155 i+(2+10 i) \sqrt {31}\right ) t+(8+175 i)}{5 \left (-2+\sqrt {31}\right )}}\right ) \cos \left (\sqrt {31} t\right )+(7128+13180 i) \sqrt {31} e^{\frac {(2+20 i) \sqrt {31} t+(8+175 i)}{5 \left (-2+\sqrt {31}\right )}}+(24800+214210 i) e^{\frac {(2+20 i) \sqrt {31} t+(8+175 i)}{5 \left (-2+\sqrt {31}\right )}}-26360 i \sqrt {31} e^{\frac {\left ((8+175 i)+20 i \sqrt {31}\right ) t+(8+40 i) \sqrt {31}}{10 \left (-2+\sqrt {31}\right )}}-49600 e^{\frac {\left ((8+175 i)+20 i \sqrt {31}\right ) t+(8+40 i) \sqrt {31}}{10 \left (-2+\sqrt {31}\right )}}+(3564-6590 i) \sqrt {31} e^{\frac {\left (175 i+2 \sqrt {31}\right ) t+20 i \sqrt {31}+(8+135 i)}{5 \left (-2+\sqrt {31}\right )}}+(12400-107105 i) e^{\frac {\left (175 i+2 \sqrt {31}\right ) t+20 i \sqrt {31}+(8+135 i)}{5 \left (-2+\sqrt {31}\right )}}-(3564-6590 i) \sqrt {31} e^{\frac {\left (20 i+(2+10 i) \sqrt {31}\right ) t+20 i \sqrt {31}+(8+135 i)}{5 \left (-2+\sqrt {31}\right )}}+(37200-321315 i) e^{\frac {\left (20 i+(2+10 i) \sqrt {31}\right ) t+20 i \sqrt {31}+(8+135 i)}{5 \left (-2+\sqrt {31}\right )}}+\left (40 \left (1240 i+659 \sqrt {31}\right ) e^{\frac {\left ((8+175 i)+20 i \sqrt {31}\right ) t+8 \sqrt {31}+620 i}{10 \left (-2+\sqrt {31}\right )}}-(691+80 i) \left (155+(10+4 i) \sqrt {31}\right ) e^{\frac {\left (175 i+2 \sqrt {31}\right ) t+20 i \sqrt {31}+(8+135 i)}{5 \left (-2+\sqrt {31}\right )}}-(691+80 i) \left (155+(10+4 i) \sqrt {31}\right ) e^{\frac {\left (20 i+(2+10 i) \sqrt {31}\right ) t+20 i \sqrt {31}+(8+135 i)}{5 \left (-2+\sqrt {31}\right )}}+(160+1382 i) \left (-155 i+(4+10 i) \sqrt {31}\right ) e^{\frac {\left (155 i+(2+10 i) \sqrt {31}\right ) t+(8+175 i)}{5 \left (-2+\sqrt {31}\right )}}\right ) \sin \left (\sqrt {31} t\right )\right ) \theta (2-t)\right )}{60001244} \]