72.21.3 problem 3

Internal problem ID [14973]
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 : 3
Date solved : Tuesday, January 28, 2025 at 07:25:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+8 y&=\left (1-\operatorname {Heaviside}\left (t -4\right )\right ) \cos \left (t -4\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 13.118 (sec). Leaf size: 145

dsolve([diff(y(t),t$2)+diff(y(t),t)+8*y(t)=(1-Heaviside(t-4))*cos(t-4),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {9 \left (\left (\sqrt {31}\, \sin \left (2 \sqrt {31}\right )-\frac {217 \cos \left (2 \sqrt {31}\right )}{9}\right ) \cos \left (\frac {\sqrt {31}\, t}{2}\right )-\frac {217 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \left (\frac {9 \sqrt {31}\, \cos \left (2 \sqrt {31}\right )}{217}+\sin \left (2 \sqrt {31}\right )\right )}{9}\right ) \operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-\frac {t}{2}+2}}{1550}-\frac {7 \left (\cos \left (4\right )-\frac {\sin \left (4\right )}{7}\right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{50}-\frac {9 \left (\cos \left (4\right )+\frac {13 \sin \left (4\right )}{9}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {31}\, \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{1550}-\frac {7 \left (-1+\operatorname {Heaviside}\left (t -4\right )\right ) \left (\left (\cos \left (t \right )+\frac {\sin \left (t \right )}{7}\right ) \cos \left (4\right )-\frac {\sin \left (4\right ) \left (\cos \left (t \right )-7 \sin \left (t \right )\right )}{7}\right )}{50} \]

✓ Solution by Mathematica

Time used: 10.293 (sec). Leaf size: 993

DSolve[{D[y[t],{t,2}]+D[y[t],t]+8*y[t]==(1-UnitStep[t-4])*Cos[t-4],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {e^{\frac {\left (-2+\sqrt {31}\right ) t+8 i \sqrt {31}+(8+124 i)}{4-2 \sqrt {31}}} \left (\left ((-7+i) \left (-98222211640+17645483561 \sqrt {31}\right ) e^{\frac {4+70 i}{-2+\sqrt {31}}}+\left ((-687555481480-158809352049 i)+(123518384927+28516125960 i) \sqrt {31}\right ) e^{\frac {(2+8 i) \sqrt {31}}{-2+\sqrt {31}}}-(7+i) \left (-98222211640+17645483561 \sqrt {31}\right ) e^{\frac {(4+54 i)+8 i \sqrt {31}}{-2+\sqrt {31}}}+\left ((-687555481480+158809352049 i)+(123518384927-28516125960 i) \sqrt {31}\right ) e^{\frac {2 \left (62 i+\sqrt {31}\right )}{-2+\sqrt {31}}}\right ) \cos \left (\frac {\sqrt {31} t}{2}\right )+\left ((9+13 i) \left (-17645483561+3168458440 \sqrt {31}\right ) e^{\frac {4+70 i}{-2+\sqrt {31}}}+\left ((158809352049-687555481480 i)-(28516125960-123518384927 i) \sqrt {31}\right ) e^{\frac {(2+8 i) \sqrt {31}}{-2+\sqrt {31}}}+(9-13 i) \left (-17645483561+3168458440 \sqrt {31}\right ) e^{\frac {(4+54 i)+8 i \sqrt {31}}{-2+\sqrt {31}}}+\left ((158809352049+687555481480 i)-(28516125960+123518384927 i) \sqrt {31}\right ) e^{\frac {2 \left (62 i+\sqrt {31}\right )}{-2+\sqrt {31}}}\right ) \sin \left (\frac {\sqrt {31} t}{2}\right )\right )}{100 \left (-98222211640+17645483561 \sqrt {31}\right )} & t>4 \\ \frac {e^{-\frac {i \left (\left (105+4 \sqrt {31}\right ) t+8 \left (2+\sqrt {31}\right )\right )}{2 \left (-2+\sqrt {31}\right )}} \left (\left (\left ((-458164195187+60587140409 i)+(82328425207-10870642399 i) \sqrt {31}\right ) e^{\frac {2 i \left (35 t+4 \sqrt {31}\right )}{-2+\sqrt {31}}}-(14-2 i) \left (-98222211640+17645483561 \sqrt {31}\right ) e^{\frac {\left ((2+105 i)-(1-4 i) \sqrt {31}\right ) t+32 i}{2 \left (-2+\sqrt {31}\right )}}-(14+2 i) \left (-98222211640+17645483561 \sqrt {31}\right ) e^{\frac {\left ((2+105 i)-(1-4 i) \sqrt {31}\right ) t+16 i \sqrt {31}}{2 \left (-2+\sqrt {31}\right )}}+\left ((-916946767773+257031563689 i)+(164708344647-46161609521 i) \sqrt {31}\right ) e^{\frac {2 i \left (\left (33+\sqrt {31}\right ) t+8\right )}{-2+\sqrt {31}}}+(7+i) \left ((-133513178762-17645483561 i)+(23982400441+3168458440 i) \sqrt {31}\right ) e^{\frac {i \left (\left (39+2 \sqrt {31}\right ) t+8 \sqrt {31}\right )}{-2+\sqrt {31}}}+\left ((-458164195187-60587140409 i)+(82328425207+10870642399 i) \sqrt {31}\right ) e^{\frac {i \left (\left (35+4 \sqrt {31}\right ) t+16\right )}{-2+\sqrt {31}}}\right ) \cos \left (\frac {\sqrt {31} t}{2}\right )+(1+i) \left (\left ((259375667798+198788527389 i)-(46599533803+35728891404 i) \sqrt {31}\right ) e^{\frac {2 i \left (35 t+4 \sqrt {31}\right )}{-2+\sqrt {31}}}+(22+4 i) \left (-17645483561+3168458440 \sqrt {31}\right ) e^{\frac {\left ((2+105 i)-(1-4 i) \sqrt {31}\right ) t+32 i}{2 \left (-2+\sqrt {31}\right )}}-(4+22 i) \left (-17645483561+3168458440 \sqrt {31}\right ) e^{\frac {\left ((2+105 i)-(1-4 i) \sqrt {31}\right ) t+16 i \sqrt {31}}{2 \left (-2+\sqrt {31}\right )}}+\left ((586989165731+329957602042 i)-(105434977084+59273367563 i) \sqrt {31}\right ) e^{\frac {2 i \left (\left (33+\sqrt {31}\right ) t+8\right )}{-2+\sqrt {31}}}+(3+4 i) \left ((-133513178762-17645483561 i)+(23982400441+3168458440 i) \sqrt {31}\right ) e^{\frac {i \left (\left (39+2 \sqrt {31}\right ) t+8 \sqrt {31}\right )}{-2+\sqrt {31}}}+(4+3 i) \left ((-62931244518-17645483561 i)+(11308566681+3168458440 i) \sqrt {31}\right ) e^{\frac {i \left (\left (35+4 \sqrt {31}\right ) t+16\right )}{-2+\sqrt {31}}}\right ) \sin \left (\frac {\sqrt {31} t}{2}\right )\right )}{200 \left (-98222211640+17645483561 \sqrt {31}\right )} & \text {True} \\ \end {array} \\ \end {array} \]