Internal problem ID [6687]
Internal file name [OUTPUT/5935_Sunday_June_05_2022_04_02_31_PM_23667262/index.tex]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 7 THE LAPLACE TRANSFORM. 7.3.1 TRANSLATION ON THE s-AXIS.
Page 297
Problem number: 70.
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, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y=1-\operatorname {Heaviside}\left (-2+t \right )-\operatorname {Heaviside}\left (t -4\right )+\operatorname {Heaviside}\left (t -6\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime \prime } + p(t)y^{\prime } + q(t) y &= F \end {align*}
Where here \begin {align*} p(t) &=4\\ q(t) &=3\\ F &=1-\operatorname {Heaviside}\left (-2+t \right )-\operatorname {Heaviside}\left (t -4\right )+\operatorname {Heaviside}\left (t -6\right ) \end {align*}
Hence the ode is \begin {align*} y^{\prime \prime }+4 y^{\prime }+3 y = 1-\operatorname {Heaviside}\left (-2+t \right )-\operatorname {Heaviside}\left (t -4\right )+\operatorname {Heaviside}\left (t -6\right ) \end {align*}
The domain of \(p(t)=4\) is \[
\{-\infty 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 )+4 s Y \left (s \right )-4 y \left (0\right )+3 Y \left (s \right ) = \frac {{\mathrm e}^{-6 s}-{\mathrm e}^{-4 s}-{\mathrm e}^{-2 s}+1}{s}\tag {1} \end {align*}
But the initial conditions are \begin {align*} y \left (0\right )&=0\\ y'(0) &=0 \end {align*}
Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )+4 s Y \left (s \right )+3 Y \left (s \right ) = \frac {{\mathrm e}^{-6 s}-{\mathrm e}^{-4 s}-{\mathrm e}^{-2 s}+1}{s} \end {align*}
Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {{\mathrm e}^{-6 s}-{\mathrm e}^{-4 s}-{\mathrm e}^{-2 s}+1}{s \left (s^{2}+4 s +3\right )} \end {align*}
Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (\frac {{\mathrm e}^{-6 s}-{\mathrm e}^{-4 s}-{\mathrm e}^{-2 s}+1}{s \left (s^{2}+4 s +3\right )}\right )\\ &= \frac {\operatorname {Heaviside}\left (2-t \right )}{3}-\frac {{\mathrm e}^{-t}}{2}+\frac {{\mathrm e}^{-3 t}}{6}+\frac {\left (-{\mathrm e}^{6-3 t}+3 \,{\mathrm e}^{2-t}\right ) \operatorname {Heaviside}\left (-2+t \right )}{6}+\frac {\left (2-3 \,{\mathrm e}^{-t +6}+{\mathrm e}^{-3 t +18}\right ) \operatorname {Heaviside}\left (t -6\right )}{6}+\frac {\left (-2-{\mathrm e}^{-3 t +12}+3 \,{\mathrm e}^{-t +4}\right ) \operatorname {Heaviside}\left (t -4\right )}{6} \end {align*}
Converting the above solution to piecewise it becomes \[
y = \left \{\begin {array}{cc} -\frac {{\mathrm e}^{-t}}{2}+\frac {{\mathrm e}^{-3 t}}{6}+\frac {1}{3} & t <2 \\ -\frac {{\mathrm e}^{-2}}{2}+\frac {{\mathrm e}^{-6}}{6}+\frac {2}{3} & t =2 \\ -\frac {{\mathrm e}^{-t}}{2}+\frac {{\mathrm e}^{-3 t}}{6}-\frac {{\mathrm e}^{6-3 t}}{6}+\frac {{\mathrm e}^{2-t}}{2} & t <4 \\ -\frac {{\mathrm e}^{-t}}{2}+\frac {{\mathrm e}^{-3 t}}{6}-\frac {{\mathrm e}^{6-3 t}}{6}+\frac {{\mathrm e}^{2-t}}{2}-\frac {1}{3}-\frac {{\mathrm e}^{-3 t +12}}{6}+\frac {{\mathrm e}^{-t +4}}{2} & t <6 \\ -\frac {{\mathrm e}^{-t}}{2}+\frac {{\mathrm e}^{-3 t}}{6}-\frac {{\mathrm e}^{6-3 t}}{6}+\frac {{\mathrm e}^{2-t}}{2}-\frac {{\mathrm e}^{-t +6}}{2}+\frac {{\mathrm e}^{-3 t +18}}{6}-\frac {{\mathrm e}^{-3 t +12}}{6}+\frac {{\mathrm e}^{-t +4}}{2} & 6\le t \end {array}\right .
\] Simplifying the solution gives \[
y = -\frac {\left (\left \{\begin {array}{cc} 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}-2 & t <2 \\ 3 \,{\mathrm e}^{-2}-{\mathrm e}^{-6}-4 & t =2 \\ 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}+{\mathrm e}^{6-3 t}-3 \,{\mathrm e}^{2-t} & t <4 \\ 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}+{\mathrm e}^{6-3 t}-3 \,{\mathrm e}^{2-t}+2+{\mathrm e}^{-3 t +12}-3 \,{\mathrm e}^{-t +4} & t <6 \\ 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}+{\mathrm e}^{6-3 t}-3 \,{\mathrm e}^{2-t}+3 \,{\mathrm e}^{-t +6}-{\mathrm e}^{-3 t +18}+{\mathrm e}^{-3 t +12}-3 \,{\mathrm e}^{-t +4} & 6\le t \end {array}\right .\right )}{6}
\]
The solution(s) found are the following \begin{align*}
\tag{1} y &= -\frac {\left (\left \{\begin {array}{cc} 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}-2 & t <2 \\ 3 \,{\mathrm e}^{-2}-{\mathrm e}^{-6}-4 & t &=2 \\ 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}+{\mathrm e}^{6-3 t}-3 \,{\mathrm e}^{2-t} & t <4 \\ 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}+{\mathrm e}^{6-3 t}-3 \,{\mathrm e}^{2-t}+2+{\mathrm e}^{-3 t +12}-3 \,{\mathrm e}^{-t +4} & t <6 \\ 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}+{\mathrm e}^{6-3 t}-3 \,{\mathrm e}^{2-t}+3 \,{\mathrm e}^{-t +6}-{\mathrm e}^{-3 t +18}+{\mathrm e}^{-3 t +12}-3 \,{\mathrm e}^{-t +4} & 6\le t \end {array}\right .\right )}{6} \\
\end{align*} Verification of solutions
\[
y = -\frac {\left (\left \{\begin {array}{cc} 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}-2 & t <2 \\ 3 \,{\mathrm e}^{-2}-{\mathrm e}^{-6}-4 & t =2 \\ 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}+{\mathrm e}^{6-3 t}-3 \,{\mathrm e}^{2-t} & t <4 \\ 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}+{\mathrm e}^{6-3 t}-3 \,{\mathrm e}^{2-t}+2+{\mathrm e}^{-3 t +12}-3 \,{\mathrm e}^{-t +4} & t <6 \\ 3 \,{\mathrm e}^{-t}-{\mathrm e}^{-3 t}+{\mathrm e}^{6-3 t}-3 \,{\mathrm e}^{2-t}+3 \,{\mathrm e}^{-t +6}-{\mathrm e}^{-3 t +18}+{\mathrm e}^{-3 t +12}-3 \,{\mathrm e}^{-t +4} & 6\le t \end {array}\right .\right )}{6}
\] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}y^{\prime }+4 y^{\prime }+3 y=1-\mathit {Heaviside}\left (-2+t \right )-\mathit {Heaviside}\left (t -4\right )+\mathit {Heaviside}\left (t -6\right ), y \left (0\right )=0, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d t}y^{\prime } \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+4 r +3=0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & \left (r +3\right ) \left (r +1\right )=0 \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-3, -1\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{-3 t} \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{-t} \\ \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=c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{-t}+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 )=1-\mathit {Heaviside}\left (-2+t \right )-\mathit {Heaviside}\left (t -4\right )+\mathit {Heaviside}\left (t -6\right )\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} {\mathrm e}^{-3 t} & {\mathrm e}^{-t} \\ -3 \,{\mathrm e}^{-3 t} & -{\mathrm e}^{-t} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=2 \,{\mathrm e}^{-4 t} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (t \right ) \\ {} & {} & y_{p}\left (t \right )=\frac {{\mathrm e}^{-3 t} \left (\int \left (-1+\mathit {Heaviside}\left (-2+t \right )+\mathit {Heaviside}\left (t -4\right )-\mathit {Heaviside}\left (t -6\right )\right ) {\mathrm e}^{3 t}d t \right )}{2}-\frac {{\mathrm e}^{-t} \left (\int \left (-1+\mathit {Heaviside}\left (-2+t \right )+\mathit {Heaviside}\left (t -4\right )-\mathit {Heaviside}\left (t -6\right )\right ) {\mathrm e}^{t}d t \right )}{2} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=-\frac {\mathit {Heaviside}\left (-2+t \right )}{3}-\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{6-3 t}}{6}-\frac {\mathit {Heaviside}\left (t -4\right )}{3}-\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{6}+\frac {\mathit {Heaviside}\left (t -6\right )}{3}+\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{6}+\frac {1}{3}+\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{2-t}}{2}+\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}-\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-t +6}}{2} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{-t}-\frac {\mathit {Heaviside}\left (-2+t \right )}{3}-\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{6-3 t}}{6}-\frac {\mathit {Heaviside}\left (t -4\right )}{3}-\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{6}+\frac {\mathit {Heaviside}\left (t -6\right )}{3}+\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{6}+\frac {1}{3}+\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{2-t}}{2}+\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}-\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-t +6}}{2} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{-t}-\frac {\mathit {Heaviside}\left (-2+t \right )}{3}-\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{6-3 t}}{6}-\frac {\mathit {Heaviside}\left (t -4\right )}{3}-\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{6}+\frac {\mathit {Heaviside}\left (t -6\right )}{3}+\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{6}+\frac {1}{3}+\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{2-t}}{2}+\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}-\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-t +6}}{2} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=c_{1} +c_{2} +\frac {1}{3} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=-3 c_{1} {\mathrm e}^{-3 t}-c_{2} {\mathrm e}^{-t}-\frac {\mathit {Dirac}\left (-2+t \right )}{3}-\frac {\mathit {Dirac}\left (-2+t \right ) {\mathrm e}^{6-3 t}}{6}+\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{6-3 t}}{2}-\frac {\mathit {Dirac}\left (t -4\right )}{3}-\frac {\mathit {Dirac}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{6}+\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{2}+\frac {\mathit {Dirac}\left (t -6\right )}{3}+\frac {\mathit {Dirac}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{6}-\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{2}+\frac {\mathit {Dirac}\left (-2+t \right ) {\mathrm e}^{2-t}}{2}-\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{2-t}}{2}+\frac {\mathit {Dirac}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}-\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}-\frac {\mathit {Dirac}\left (t -6\right ) {\mathrm e}^{-t +6}}{2}+\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-t +6}}{2} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=-3 c_{1} -c_{2} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =\frac {1}{6}, c_{2} =-\frac {1}{2}\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{-3 t}}{6}-\frac {{\mathrm e}^{-t}}{2}-\frac {\mathit {Heaviside}\left (-2+t \right )}{3}-\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{6-3 t}}{6}-\frac {\mathit {Heaviside}\left (t -4\right )}{3}-\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{6}+\frac {\mathit {Heaviside}\left (t -6\right )}{3}+\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{6}+\frac {1}{3}+\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{2-t}}{2}+\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}-\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-t +6}}{2} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{-3 t}}{6}-\frac {{\mathrm e}^{-t}}{2}-\frac {\mathit {Heaviside}\left (-2+t \right )}{3}-\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{6-3 t}}{6}-\frac {\mathit {Heaviside}\left (t -4\right )}{3}-\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{6}+\frac {\mathit {Heaviside}\left (t -6\right )}{3}+\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{6}+\frac {1}{3}+\frac {\mathit {Heaviside}\left (-2+t \right ) {\mathrm e}^{2-t}}{2}+\frac {\mathit {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}-\frac {\mathit {Heaviside}\left (t -6\right ) {\mathrm e}^{-t +6}}{2} \end {array} \]
Maple trace
✓ Solution by Maple
Time used: 2.109 (sec). Leaf size: 94
\[
y \left (t \right ) = \frac {1}{3}-\frac {\operatorname {Heaviside}\left (t -4\right )}{3}+\frac {{\mathrm e}^{-3 t}}{6}-\frac {{\mathrm e}^{-t}}{2}-\frac {\operatorname {Heaviside}\left (t -6\right ) {\mathrm e}^{-t +6}}{2}+\frac {\operatorname {Heaviside}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{6}+\frac {\operatorname {Heaviside}\left (t -6\right )}{3}+\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}-\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{6}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}-\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}-\frac {\operatorname {Heaviside}\left (t -2\right )}{3}
\]
✓ Solution by Mathematica
Time used: 0.059 (sec). Leaf size: 175
\[
y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{6} e^{-3 t} \left (-1+e^t\right )^2 \left (1+2 e^t\right ) & t\leq 2 \\ -\frac {1}{6} e^{-3 t} \left (-1+e^2\right ) \left (1+e^2+e^4-3 e^{2 t}\right ) & 26.20.2 Maple step by step solution
`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`
dsolve([diff(y(t),t$2)+4*diff(y(t),t)+3*y(t)=1-Heaviside(t-2)-Heaviside(t-4)+Heaviside(t-6),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
DSolve[{y''[t]+4*y'[t]+3*y[t]==1-UnitStep[t-2]-UnitStep[t-4]+UnitStep[t-6],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]