Internal problem ID [6506]
Internal file name [OUTPUT/5754_Sunday_June_05_2022_03_52_52_PM_88188911/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: 3(b).
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]]
\[ \boxed {y^{\prime \prime }+3 y^{\prime }-2 y=-6 \,{\mathrm e}^{\pi -t}} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 1, y^{\prime }\left (\pi \right ) = 4] \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) &=3\\ q(t) &=-2\\ F &=-6 \,{\mathrm e}^{\pi -t} \end {align*}
Hence the ode is \begin {align*} y^{\prime \prime }+3 y^{\prime }-2 y = -6 \,{\mathrm e}^{\pi -t} \end {align*}
The domain of \(p(t)=3\) is \[
\{-\infty Since both initial conditions are not at zero, then let \begin {align*} y(0) &= 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 )+3 s Y \left (s \right )-3 y \left (0\right )-2 Y \left (s \right ) = -\frac {6 \,{\mathrm e}^{\pi }}{s +1}\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} +3 s Y \left (s \right )-3 c_{1} -2 Y \left (s \right ) = -\frac {6 \,{\mathrm e}^{\pi }}{s +1} \end {align*}
Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = -\frac {-s^{2} c_{1} -4 s c_{1} -c_{2} s +6 \,{\mathrm e}^{\pi }-3 c_{1} -c_{2}}{\left (s +1\right ) \left (s^{2}+3 s -2\right )} \end {align*}
Applying partial fractions decomposition results in \[ Y(s)= \frac {\left (-\frac {3 \,{\mathrm e}^{\pi }}{34}+\frac {3 c_{1}}{17}+\frac {2 c_{2}}{17}\right ) \left (-\frac {3}{2}+\frac {\sqrt {17}}{2}\right )-\frac {15 \,{\mathrm e}^{\pi }}{17}+\frac {13 c_{1}}{17}+\frac {3 c_{2}}{17}}{s +\frac {3}{2}-\frac {\sqrt {17}}{2}}+\frac {\left (-\frac {3 \,{\mathrm e}^{\pi }}{34}+\frac {3 c_{1}}{17}+\frac {2 c_{2}}{17}\right ) \left (-\frac {3}{2}-\frac {\sqrt {17}}{2}\right )-\frac {15 \,{\mathrm e}^{\pi }}{17}+\frac {13 c_{1}}{17}+\frac {3 c_{2}}{17}}{s +\frac {3}{2}+\frac {\sqrt {17}}{2}}+\frac {3 \,{\mathrm e}^{\pi }}{2 \left (s +1\right )} \] The inverse Laplace of each term above
is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (\frac {\left (-\frac {3 \,{\mathrm e}^{\pi }}{34}+\frac {3 c_{1}}{17}+\frac {2 c_{2}}{17}\right ) \left (-\frac {3}{2}+\frac {\sqrt {17}}{2}\right )-\frac {15 \,{\mathrm e}^{\pi }}{17}+\frac {13 c_{1}}{17}+\frac {3 c_{2}}{17}}{s +\frac {3}{2}-\frac {\sqrt {17}}{2}}\right ) &= \frac {{\mathrm e}^{-\frac {3 t}{2}+\frac {t \sqrt {17}}{2}} \left (4 c_{2} \sqrt {17}-3 \,{\mathrm e}^{\pi } \sqrt {17}-51 \,{\mathrm e}^{\pi }+2 c_{1} \left (17+3 \sqrt {17}\right )\right )}{68}\\ \mathcal {L}^{-1}\left (\frac {\left (-\frac {3 \,{\mathrm e}^{\pi }}{34}+\frac {3 c_{1}}{17}+\frac {2 c_{2}}{17}\right ) \left (-\frac {3}{2}-\frac {\sqrt {17}}{2}\right )-\frac {15 \,{\mathrm e}^{\pi }}{17}+\frac {13 c_{1}}{17}+\frac {3 c_{2}}{17}}{s +\frac {3}{2}+\frac {\sqrt {17}}{2}}\right ) &= \frac {{\mathrm e}^{\frac {\left (-3-\sqrt {17}\right ) \left (-\frac {3 \left (3+\sqrt {17}\right ) t}{2 \left (-\frac {3}{2}-\frac {\sqrt {17}}{2}\right )}-\frac {\left (3+\sqrt {17}\right ) t \sqrt {17}}{2 \left (-\frac {3}{2}-\frac {\sqrt {17}}{2}\right )}\right )}{6+2 \sqrt {17}}} \left (-4 c_{2} \sqrt {17}+3 \,{\mathrm e}^{\pi } \sqrt {17}-51 \,{\mathrm e}^{\pi }-2 c_{1} \left (-17+3 \sqrt {17}\right )\right )}{68}\\ \mathcal {L}^{-1}\left (\frac {3 \,{\mathrm e}^{\pi }}{2 \left (s +1\right )}\right ) &= \frac {3 \,{\mathrm e}^{\pi -t}}{2} \end {align*}
Adding the above results and simplifying gives \[ y=\frac {3 \,{\mathrm e}^{\pi -t}}{2}+\frac {\left (17 \cosh \left (\frac {t \sqrt {17}}{2}\right ) \left (-3 \,{\mathrm e}^{\pi }+2 c_{1} \right )+\sqrt {17}\, \sinh \left (\frac {t \sqrt {17}}{2}\right ) \left (-3 \,{\mathrm e}^{\pi }+6 c_{1} +4 c_{2} \right )\right ) {\mathrm e}^{-\frac {3 t}{2}}}{34} \] Since both initial conditions given are not at
zero, then we need to setup two equations to solve for \(c_{1},c_{1}\). At \(t=\pi \) the first equation becomes, using
the above solution \begin {align*} 1 &= \frac {3}{2}+\frac {\left (17 \cosh \left (\frac {\pi \sqrt {17}}{2}\right ) \left (-3 \,{\mathrm e}^{\pi }+2 c_{1} \right )+\sqrt {17}\, \sinh \left (\frac {\pi \sqrt {17}}{2}\right ) \left (-3 \,{\mathrm e}^{\pi }+6 c_{1} +4 c_{2} \right )\right ) {\mathrm e}^{-\frac {3 \pi }{2}}}{34} \end {align*}
And taking derivative of the solution and evaluating at \(t=\pi \) gives the second equation as
\begin {align*} 4 &= -\frac {3}{2}+\frac {\left (\frac {17 \sqrt {17}\, \sinh \left (\frac {\pi \sqrt {17}}{2}\right ) \left (-3 \,{\mathrm e}^{\pi }+2 c_{1} \right )}{2}+\frac {17 \cosh \left (\frac {\pi \sqrt {17}}{2}\right ) \left (-3 \,{\mathrm e}^{\pi }+6 c_{1} +4 c_{2} \right )}{2}\right ) {\mathrm e}^{-\frac {3 \pi }{2}}}{34}-\frac {3 \left (17 \cosh \left (\frac {\pi \sqrt {17}}{2}\right ) \left (-3 \,{\mathrm e}^{\pi }+2 c_{1} \right )+\sqrt {17}\, \sinh \left (\frac {\pi \sqrt {17}}{2}\right ) \left (-3 \,{\mathrm e}^{\pi }+6 c_{1} +4 c_{2} \right )\right ) {\mathrm e}^{-\frac {3 \pi }{2}}}{68} \end {align*}
Solving gives \begin {align*} c_{1} &= \frac {{\mathrm e}^{\frac {3 \pi }{2}} \left (51 \,{\mathrm e}^{\pi } \sinh \left (\frac {\pi \sqrt {17}}{2}\right )^{2} {\mathrm e}^{-\frac {3 \pi }{2}}-51 \,{\mathrm e}^{\pi } {\mathrm e}^{-\frac {3 \pi }{2}} \cosh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}+19 \sqrt {17}\, \sinh \left (\frac {\pi \sqrt {17}}{2}\right )+17 \cosh \left (\frac {\pi \sqrt {17}}{2}\right )\right )}{34 \sinh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}-34 \cosh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}}\\ c_{2} &= -\frac {\left (51 \,{\mathrm e}^{\pi } \sinh \left (\frac {\pi \sqrt {17}}{2}\right )^{2} {\mathrm e}^{-\frac {3 \pi }{2}}-51 \,{\mathrm e}^{\pi } {\mathrm e}^{-\frac {3 \pi }{2}} \cosh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}+37 \sqrt {17}\, \sinh \left (\frac {\pi \sqrt {17}}{2}\right )+187 \cosh \left (\frac {\pi \sqrt {17}}{2}\right )\right ) {\mathrm e}^{\frac {3 \pi }{2}}}{2 \left (17 \sinh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}-17 \cosh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}\right )} \end {align*}
Subtituting these in the solution obtained above gives \begin {align*} y &= \frac {3 \,{\mathrm e}^{\pi -t}}{2}+\frac {\left (17 \cosh \left (\frac {t \sqrt {17}}{2}\right ) \left (-3 \,{\mathrm e}^{\pi }+\frac {{\mathrm e}^{\frac {3 \pi }{2}} \left (51 \,{\mathrm e}^{\pi } \sinh \left (\frac {\pi \sqrt {17}}{2}\right )^{2} {\mathrm e}^{-\frac {3 \pi }{2}}-51 \,{\mathrm e}^{\pi } {\mathrm e}^{-\frac {3 \pi }{2}} \cosh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}+19 \sqrt {17}\, \sinh \left (\frac {\pi \sqrt {17}}{2}\right )+17 \cosh \left (\frac {\pi \sqrt {17}}{2}\right )\right )}{17 \sinh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}-17 \cosh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}}\right )+\sqrt {17}\, \sinh \left (\frac {t \sqrt {17}}{2}\right ) \left (-3 \,{\mathrm e}^{\pi }+\frac {3 \,{\mathrm e}^{\frac {3 \pi }{2}} \left (51 \,{\mathrm e}^{\pi } \sinh \left (\frac {\pi \sqrt {17}}{2}\right )^{2} {\mathrm e}^{-\frac {3 \pi }{2}}-51 \,{\mathrm e}^{\pi } {\mathrm e}^{-\frac {3 \pi }{2}} \cosh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}+19 \sqrt {17}\, \sinh \left (\frac {\pi \sqrt {17}}{2}\right )+17 \cosh \left (\frac {\pi \sqrt {17}}{2}\right )\right )}{17 \sinh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}-17 \cosh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}}-\frac {2 \left (51 \,{\mathrm e}^{\pi } \sinh \left (\frac {\pi \sqrt {17}}{2}\right )^{2} {\mathrm e}^{-\frac {3 \pi }{2}}-51 \,{\mathrm e}^{\pi } {\mathrm e}^{-\frac {3 \pi }{2}} \cosh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}+37 \sqrt {17}\, \sinh \left (\frac {\pi \sqrt {17}}{2}\right )+187 \cosh \left (\frac {\pi \sqrt {17}}{2}\right )\right ) {\mathrm e}^{\frac {3 \pi }{2}}}{17 \sinh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}-17 \cosh \left (\frac {\pi \sqrt {17}}{2}\right )^{2}}\right )\right ) {\mathrm e}^{-\frac {3 t}{2}}}{34}\\ &= -\frac {19 \left (-\frac {51 \,{\mathrm e}^{-\frac {\pi }{2}-t}}{19}+{\mathrm e}^{-\frac {3 t}{2}} \left (\left (-\sqrt {17}\, \sinh \left (\frac {t \sqrt {17}}{2}\right )+\frac {17 \cosh \left (\frac {t \sqrt {17}}{2}\right )}{19}\right ) \cosh \left (\frac {\pi \sqrt {17}}{2}\right )+\sinh \left (\frac {\pi \sqrt {17}}{2}\right ) \left (\sqrt {17}\, \cosh \left (\frac {t \sqrt {17}}{2}\right )-\frac {17 \sinh \left (\frac {t \sqrt {17}}{2}\right )}{19}\right )\right )\right ) {\mathrm e}^{\frac {3 \pi }{2}}}{34} \end {align*}
Simplifying the solution gives \[
y = -\frac {19 \left (-\frac {51 \,{\mathrm e}^{-\frac {\pi }{2}-t}}{19}+{\mathrm e}^{-\frac {3 t}{2}} \left (\left (-\sqrt {17}\, \sinh \left (\frac {t \sqrt {17}}{2}\right )+\frac {17 \cosh \left (\frac {t \sqrt {17}}{2}\right )}{19}\right ) \cosh \left (\frac {\pi \sqrt {17}}{2}\right )+\sinh \left (\frac {\pi \sqrt {17}}{2}\right ) \left (\sqrt {17}\, \cosh \left (\frac {t \sqrt {17}}{2}\right )-\frac {17 \sinh \left (\frac {t \sqrt {17}}{2}\right )}{19}\right )\right )\right ) {\mathrm e}^{\frac {3 \pi }{2}}}{34}
\] The solution(s) found are the following \begin{align*}
\tag{1} y &= -\frac {19 \left (-\frac {51 \,{\mathrm e}^{-\frac {\pi }{2}-t}}{19}+{\mathrm e}^{-\frac {3 t}{2}} \left (\left (-\sqrt {17}\, \sinh \left (\frac {t \sqrt {17}}{2}\right )+\frac {17 \cosh \left (\frac {t \sqrt {17}}{2}\right )}{19}\right ) \cosh \left (\frac {\pi \sqrt {17}}{2}\right )+\sinh \left (\frac {\pi \sqrt {17}}{2}\right ) \left (\sqrt {17}\, \cosh \left (\frac {t \sqrt {17}}{2}\right )-\frac {17 \sinh \left (\frac {t \sqrt {17}}{2}\right )}{19}\right )\right )\right ) {\mathrm e}^{\frac {3 \pi }{2}}}{34} \\
\end{align*} Verification of solutions
\[
y = -\frac {19 \left (-\frac {51 \,{\mathrm e}^{-\frac {\pi }{2}-t}}{19}+{\mathrm e}^{-\frac {3 t}{2}} \left (\left (-\sqrt {17}\, \sinh \left (\frac {t \sqrt {17}}{2}\right )+\frac {17 \cosh \left (\frac {t \sqrt {17}}{2}\right )}{19}\right ) \cosh \left (\frac {\pi \sqrt {17}}{2}\right )+\sinh \left (\frac {\pi \sqrt {17}}{2}\right ) \left (\sqrt {17}\, \cosh \left (\frac {t \sqrt {17}}{2}\right )-\frac {17 \sinh \left (\frac {t \sqrt {17}}{2}\right )}{19}\right )\right )\right ) {\mathrm e}^{\frac {3 \pi }{2}}}{34}
\] Verified OK. Maple trace
✓ Solution by Maple
Time used: 2.078 (sec). Leaf size: 53
\[
y \left (t \right ) = -\frac {19 \sinh \left (\frac {\left (\pi -t \right ) \sqrt {17}}{2}\right ) \sqrt {17}\, {\mathrm e}^{-\frac {3 t}{2}+\frac {3 \pi }{2}}}{34}-\frac {\cosh \left (\frac {\left (\pi -t \right ) \sqrt {17}}{2}\right ) {\mathrm e}^{-\frac {3 t}{2}+\frac {3 \pi }{2}}}{2}+\frac {3 \,{\mathrm e}^{\pi -t}}{2}
\]
✓ Solution by Mathematica
Time used: 0.354 (sec). Leaf size: 103
\[
y(t)\to \frac {1}{68} e^{-\frac {1}{2} \left (3+\sqrt {17}\right ) t-\frac {1}{2} \left (\sqrt {17}-3\right ) \pi } \left (\left (19 \sqrt {17}-17\right ) e^{\sqrt {17} t}+102 e^{\frac {1}{2} \left (\left (1+\sqrt {17}\right ) t+\left (\sqrt {17}-1\right ) \pi \right )}-\left (\left (17+19 \sqrt {17}\right ) e^{\sqrt {17} \pi }\right )\right )
\]
`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)+3*diff(y(t),t)-2*y(t)=-6*exp(Pi-t),y(Pi) = 1, D(y)(Pi) = 4],y(t), singsol=all)
DSolve[{y''[t]+3*y'[t]-2*y[t]==-6*Exp[Pi-t],{y[Pi]==1,y'[Pi]==4}},y[t],t,IncludeSingularSolutions -> True]