Internal problem ID [6511]
Internal file name [OUTPUT/5759_Sunday_June_05_2022_03_53_02_PM_19961650/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 7. Laplace Transforms. Section 7.5 Problesm for review and discovery. Section
A, Drill exercises. Page 309
Problem number: 4(c).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_laplace", "second_order_linear_constant_coeff"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+y^{\prime }+2 y=t} \] Since no initial conditions are explicitly given, then let \begin {align*} y \left (0\right )&= c_{1} \\ y'(0) &= c_{2} \end {align*}
Solving using the Laplace transform method. Let \begin {align*} \mathcal {L}\left (y\right ) =Y(s) \end {align*}
Taking the Laplace transform of the ode and using the relations that \begin {align*} \mathcal {L}\left (y^{\prime }\right ) &= s Y(s) - y \left (0\right )\\ \mathcal {L}\left (y^{\prime \prime }\right ) &= s^2 Y(s) - y'(0) - s y \left (0\right ) \end {align*}
The given ode now becomes an algebraic equation in the Laplace domain \begin {align*} s^{2} Y \left (s \right )-y^{\prime }\left (0\right )-s y \left (0\right )+s Y \left (s \right )-y \left (0\right )+2 Y \left (s \right ) = \frac {1}{s^{2}}\tag {1} \end {align*}
But the initial conditions are \begin {align*} y \left (0\right )&=c_{1}\\ y'(0) &=c_{2} \end {align*}
Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )-c_{2} -s c_{1} +s Y \left (s \right )-c_{1} +2 Y \left (s \right ) = \frac {1}{s^{2}} \end {align*}
Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {c_{1} s^{3}+c_{1} s^{2}+c_{2} s^{2}+1}{s^{2} \left (s^{2}+s +2\right )} \end {align*}
Applying partial fractions decomposition results in \[ Y(s)= -\frac {1}{4 s}+\frac {1}{2 s^{2}}+\frac {\left (-\frac {c_{1}}{7}-\frac {2 c_{2}}{7}+\frac {3}{28}\right ) \left (-\frac {1}{2}+\frac {i \sqrt {7}}{2}\right )+\frac {3 c_{1}}{7}-\frac {c_{2}}{7}+\frac {5}{28}}{s +\frac {1}{2}-\frac {i \sqrt {7}}{2}}+\frac {\left (-\frac {c_{1}}{7}-\frac {2 c_{2}}{7}+\frac {3}{28}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {7}}{2}\right )+\frac {3 c_{1}}{7}-\frac {c_{2}}{7}+\frac {5}{28}}{s +\frac {1}{2}+\frac {i \sqrt {7}}{2}} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (-\frac {1}{4 s}\right ) &= -{\frac {1}{4}}\\ \mathcal {L}^{-1}\left (\frac {1}{2 s^{2}}\right ) &= \frac {t}{2}\\ \mathcal {L}^{-1}\left (\frac {\left (-\frac {c_{1}}{7}-\frac {2 c_{2}}{7}+\frac {3}{28}\right ) \left (-\frac {1}{2}+\frac {i \sqrt {7}}{2}\right )+\frac {3 c_{1}}{7}-\frac {c_{2}}{7}+\frac {5}{28}}{s +\frac {1}{2}-\frac {i \sqrt {7}}{2}}\right ) &= \frac {{\mathrm e}^{-\frac {\left (1-i \sqrt {7}\right ) \left (-\frac {t}{2 \left (-\frac {1}{2}+\frac {i \sqrt {7}}{2}\right )}+\frac {i \sqrt {7}\, t}{-1+i \sqrt {7}}\right )}{2}} \left (7-8 i \sqrt {7}\, c_{2} +3 i \sqrt {7}+4 \left (-i \sqrt {7}+7\right ) c_{1} \right )}{56}\\ \mathcal {L}^{-1}\left (\frac {\left (-\frac {c_{1}}{7}-\frac {2 c_{2}}{7}+\frac {3}{28}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {7}}{2}\right )+\frac {3 c_{1}}{7}-\frac {c_{2}}{7}+\frac {5}{28}}{s +\frac {1}{2}+\frac {i \sqrt {7}}{2}}\right ) &= \frac {{\mathrm e}^{-\frac {\left (1+i \sqrt {7}\right ) \left (-\frac {t}{2 \left (-\frac {1}{2}-\frac {i \sqrt {7}}{2}\right )}-\frac {i \sqrt {7}\, t}{2 \left (-\frac {1}{2}-\frac {i \sqrt {7}}{2}\right )}\right )}{2}} \left (7+8 i \sqrt {7}\, c_{2} -3 i \sqrt {7}+4 c_{1} \left (i \sqrt {7}+7\right )\right )}{56} \end {align*}
Adding the above results and simplifying gives \[ y=-\frac {1}{4}+\frac {t}{2}+\frac {\left (7 \cos \left (\frac {\sqrt {7}\, t}{2}\right ) \left (1+4 c_{1} \right )+\sin \left (\frac {\sqrt {7}\, t}{2}\right ) \sqrt {7}\, \left (4 c_{1} +8 c_{2} -3\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{28} \] Simplifying the solution gives \[ y = -\frac {1}{4}+\frac {\sqrt {7}\, \left (c_{1} +2 c_{2} -\frac {3}{4}\right ) {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right )}{7}+\frac {\left (1+4 c_{1} \right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right )}{4}+\frac {t}{2} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {1}{4}+\frac {\sqrt {7}\, \left (c_{1} +2 c_{2} -\frac {3}{4}\right ) {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right )}{7}+\frac {\left (1+4 c_{1} \right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right )}{4}+\frac {t}{2} \\ \end{align*}
Verification of solutions
\[ y = -\frac {1}{4}+\frac {\sqrt {7}\, \left (c_{1} +2 c_{2} -\frac {3}{4}\right ) {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right )}{7}+\frac {\left (1+4 c_{1} \right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right )}{4}+\frac {t}{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime }+y^{\prime }+2 y=t \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+r +2=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-1\right )\pm \left (\sqrt {-7}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}, -\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )=\cos \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )=\sin \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y=c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right )+y_{p}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y=\cos \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} c_{1} +\sin \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} c_{2} +y_{p}\left (t \right ) \\ \square & {} & \textrm {Find a particular solution}\hspace {3pt} y_{p}\left (t \right )\hspace {3pt}\textrm {of the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Use variation of parameters to find}\hspace {3pt} y_{p}\hspace {3pt}\textrm {here}\hspace {3pt} f \left (t \right )\hspace {3pt}\textrm {is the forcing function}\hspace {3pt} \\ {} & {} & \left [y_{p}\left (t \right )=-y_{1}\left (t \right ) \left (\int \frac {y_{2}\left (t \right ) f \left (t \right )}{W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )}d t \right )+y_{2}\left (t \right ) \left (\int \frac {y_{1}\left (t \right ) f \left (t \right )}{W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )}d t \right ), f \left (t \right )=t \right ] \\ {} & \circ & \textrm {Wronskian of solutions of the homogeneous equation}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=\left [\begin {array}{cc} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} & \sin \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \\ -\frac {\sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{2}-\frac {\cos \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{2} & \frac {\sqrt {7}\, \cos \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{2}-\frac {\sin \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{2} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=\frac {\sqrt {7}\, {\mathrm e}^{-t}}{2} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (t \right ) \\ {} & {} & y_{p}\left (t \right )=-\frac {2 \sqrt {7}\, {\mathrm e}^{-\frac {t}{2}} \left (\cos \left (\frac {\sqrt {7}\, t}{2}\right ) \left (\int t \,{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right )d t \right )-\sin \left (\frac {\sqrt {7}\, t}{2}\right ) \left (\int t \,{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right )d t \right )\right )}{7} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=\frac {t}{2}-\frac {1}{4} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=\cos \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} c_{1} +\sin \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} c_{2} +\frac {t}{2}-\frac {1}{4} \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients <- constant coefficients successful <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 2.0 (sec). Leaf size: 52
dsolve(diff(y(t),t$2)+diff(y(t),t)+2*y(t)=t,y(t), singsol=all)
\[ y \left (t \right ) = -\frac {1}{4}+\frac {t}{2}+\frac {\left (7 \cos \left (\frac {\sqrt {7}\, t}{2}\right ) \left (1+4 y \left (0\right )\right )+\sin \left (\frac {\sqrt {7}\, t}{2}\right ) \sqrt {7}\, \left (8 D\left (y \right )\left (0\right )+4 y \left (0\right )-3\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{28} \]
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 56
DSolve[y''[t]+y'[t]+2*y[t]==t,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {t}{2}+c_2 e^{-t/2} \cos \left (\frac {\sqrt {7} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {7} t}{2}\right )-\frac {1}{4} \]