Internal problem ID [14663]
Internal file name [OUTPUT/14343_Wednesday_April_03_2024_02_17_24_PM_73571264/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.5, page 175
Problem number: 45.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "higher_order_linear_constant_coefficients_ODE"
Maple gives the following as the ode type
[[_high_order, _missing_x]]
\[ \boxed {y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 3, y^{\prime \prime }\left (0\right ) = -7, y^{\prime \prime \prime }\left (0\right ) = 15] \end {align*}
The characteristic equation is \[ \lambda ^{4}-5 \lambda ^{2}+4 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 2\\ \lambda _2 &= -2\\ \lambda _3 &= 1\\ \lambda _4 &= -1 \end {align*}
Therefore the homogeneous solution is \[ y_h(t)=c_{1} {\mathrm e}^{-t}+c_{2} {\mathrm e}^{-2 t}+c_{3} {\mathrm e}^{t}+c_{4} {\mathrm e}^{2 t} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-t}\\ y_2 &= {\mathrm e}^{-2 t}\\ y_3 &= {\mathrm e}^{t}\\ y_4 &= {\mathrm e}^{2 t} \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = c_{1} {\mathrm e}^{-t}+c_{2} {\mathrm e}^{-2 t}+c_{3} {\mathrm e}^{t}+c_{4} {\mathrm e}^{2 t} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = -1\) and \(t = 0\) in the above gives \begin {align*} -1 = c_{1} +c_{2} +c_{3} +c_{4}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = -c_{1} {\mathrm e}^{-t}-2 c_{2} {\mathrm e}^{-2 t}+c_{3} {\mathrm e}^{t}+2 c_{4} {\mathrm e}^{2 t} \end {align*}
substituting \(y^{\prime } = 3\) and \(t = 0\) in the above gives \begin {align*} 3 = -c_{1} -2 c_{2} +c_{3} +2 c_{4}\tag {2A} \end {align*}
Taking two derivatives of the solution gives \begin {align*} y^{\prime \prime } = c_{1} {\mathrm e}^{-t}+4 c_{2} {\mathrm e}^{-2 t}+c_{3} {\mathrm e}^{t}+4 c_{4} {\mathrm e}^{2 t} \end {align*}
substituting \(y^{\prime \prime } = -7\) and \(t = 0\) in the above gives \begin {align*} -7 = c_{1} +4 c_{2} +c_{3} +4 c_{4}\tag {3A} \end {align*}
Taking three derivatives of the solution gives \begin {align*} y^{\prime \prime \prime } = -c_{1} {\mathrm e}^{-t}-8 c_{2} {\mathrm e}^{-2 t}+c_{3} {\mathrm e}^{t}+8 c_{4} {\mathrm e}^{2 t} \end {align*}
substituting \(y^{\prime \prime \prime } = 15\) and \(t = 0\) in the above gives \begin {align*} 15 = -c_{1} -8 c_{2} +c_{3} +8 c_{4}\tag {4A} \end {align*}
Equations {1A,2A,3A,4A} are now solved for \(\{c_{1}, c_{2}, c_{3}, c_{4}\}\). Solving for the constants gives \begin {align*} c_{1}&=1\\ c_{2}&=-2\\ c_{3}&=0\\ c_{4}&=0 \end {align*}
Substituting these values back in above solution results in \begin {align*} y = {\mathrm e}^{-t}-2 \,{\mathrm e}^{-2 t} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-t}-2 \,{\mathrm e}^{-2 t} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-t}-2 \,{\mathrm e}^{-2 t} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y=0, y \left (0\right )=-1, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=3, y^{\prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=-7, y^{\prime \prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=15\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & y^{\prime \prime \prime \prime } \\ \square & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{1}\left (t \right ) \\ {} & {} & y_{1}\left (t \right )=y \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (t \right ) \\ {} & {} & y_{2}\left (t \right )=y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (t \right ) \\ {} & {} & y_{3}\left (t \right )=y^{\prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{4}\left (t \right ) \\ {} & {} & y_{4}\left (t \right )=y^{\prime \prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{4}^{\prime }\left (t \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{4}^{\prime }\left (t \right )=5 y_{3}\left (t \right )-4 y_{1}\left (t \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (t \right )=y_{1}^{\prime }\left (t \right ), y_{3}\left (t \right )=y_{2}^{\prime }\left (t \right ), y_{4}\left (t \right )=y_{3}^{\prime }\left (t \right ), y_{4}^{\prime }\left (t \right )=5 y_{3}\left (t \right )-4 y_{1}\left (t \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (t \right )=\left [\begin {array}{c} y_{1}\left (t \right ) \\ y_{2}\left (t \right ) \\ y_{3}\left (t \right ) \\ y_{4}\left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (t \right )=\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -4 & 0 & 5 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (t \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -4 & 0 & 5 & 0 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (t \right )=A \cdot {\moverset {\rightarrow }{y}}\left (t \right ) \\ \bullet & {} & \textrm {To solve the system, find the eigenvalues and eigenvectors of}\hspace {3pt} A \\ \bullet & {} & \textrm {Eigenpairs of}\hspace {3pt} A \\ {} & {} & \left [\left [-2, \left [\begin {array}{c} -\frac {1}{8} \\ \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]\right ], \left [-1, \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ]\right ], \left [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ]\right ], \left [2, \left [\begin {array}{c} \frac {1}{8} \\ \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-2, \left [\begin {array}{c} -\frac {1}{8} \\ \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{-2 t}\cdot \left [\begin {array}{c} -\frac {1}{8} \\ \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-1, \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}={\mathrm e}^{-t}\cdot \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}={\mathrm e}^{t}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [2, \left [\begin {array}{c} \frac {1}{8} \\ \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{4}={\mathrm e}^{2 t}\cdot \left [\begin {array}{c} \frac {1}{8} \\ \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {General solution to the system of ODEs}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\moverset {\rightarrow }{y}}_{1}+c_{2} {\moverset {\rightarrow }{y}}_{2}+c_{3} {\moverset {\rightarrow }{y}}_{3}+c_{4} {\moverset {\rightarrow }{y}}_{4} \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\mathrm e}^{-2 t}\cdot \left [\begin {array}{c} -\frac {1}{8} \\ \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{-t}\cdot \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ]+c_{3} {\mathrm e}^{t}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ]+c_{4} {\mathrm e}^{2 t}\cdot \left [\begin {array}{c} \frac {1}{8} \\ \frac {1}{4} \\ \frac {1}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=-\frac {\left (-{\mathrm e}^{4 t} c_{4} -8 c_{3} {\mathrm e}^{3 t}+8 c_{2} {\mathrm e}^{t}+c_{1} \right ) {\mathrm e}^{-2 t}}{8} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y \left (0\right )=-1 \\ {} & {} & -1=\frac {c_{4}}{8}+c_{3} -c_{2} -\frac {c_{1}}{8} \\ \bullet & {} & \textrm {Calculate the 1st derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\left (-4 \,{\mathrm e}^{4 t} c_{4} -24 c_{3} {\mathrm e}^{3 t}+8 c_{2} {\mathrm e}^{t}\right ) {\mathrm e}^{-2 t}}{8}+\frac {\left (-{\mathrm e}^{4 t} c_{4} -8 c_{3} {\mathrm e}^{3 t}+8 c_{2} {\mathrm e}^{t}+c_{1} \right ) {\mathrm e}^{-2 t}}{4} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=3 \\ {} & {} & 3=\frac {c_{4}}{4}+c_{3} +c_{2} +\frac {c_{1}}{4} \\ \bullet & {} & \textrm {Calculate the 2nd derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=-\frac {\left (-16 \,{\mathrm e}^{4 t} c_{4} -72 c_{3} {\mathrm e}^{3 t}+8 c_{2} {\mathrm e}^{t}\right ) {\mathrm e}^{-2 t}}{8}+\frac {\left (-4 \,{\mathrm e}^{4 t} c_{4} -24 c_{3} {\mathrm e}^{3 t}+8 c_{2} {\mathrm e}^{t}\right ) {\mathrm e}^{-2 t}}{2}-\frac {\left (-{\mathrm e}^{4 t} c_{4} -8 c_{3} {\mathrm e}^{3 t}+8 c_{2} {\mathrm e}^{t}+c_{1} \right ) {\mathrm e}^{-2 t}}{2} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=-7 \\ {} & {} & -7=\frac {c_{4}}{2}+c_{3} -c_{2} -\frac {c_{1}}{2} \\ \bullet & {} & \textrm {Calculate the 3rd derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }=-\frac {\left (-64 \,{\mathrm e}^{4 t} c_{4} -216 c_{3} {\mathrm e}^{3 t}+8 c_{2} {\mathrm e}^{t}\right ) {\mathrm e}^{-2 t}}{8}+\frac {3 \left (-16 \,{\mathrm e}^{4 t} c_{4} -72 c_{3} {\mathrm e}^{3 t}+8 c_{2} {\mathrm e}^{t}\right ) {\mathrm e}^{-2 t}}{4}-\frac {3 \left (-4 \,{\mathrm e}^{4 t} c_{4} -24 c_{3} {\mathrm e}^{3 t}+8 c_{2} {\mathrm e}^{t}\right ) {\mathrm e}^{-2 t}}{2}+\left (-{\mathrm e}^{4 t} c_{4} -8 c_{3} {\mathrm e}^{3 t}+8 c_{2} {\mathrm e}^{t}+c_{1} \right ) {\mathrm e}^{-2 t} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime \prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=15 \\ {} & {} & 15=c_{1} +c_{2} +c_{3} +c_{4} \\ \bullet & {} & \textrm {Solve for the unknown coefficients}\hspace {3pt} \\ {} & {} & \left \{c_{1} =16, c_{2} =-1, c_{3} =0, c_{4} =0\right \} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\left ({\mathrm e}^{t}-2\right ) {\mathrm e}^{-2 t} \end {array} \]
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 15
dsolve([diff(y(t),t$4)-5*diff(y(t),t$2)+4*y(t)=0,y(0) = -1, D(y)(0) = 3, (D@@2)(y)(0) = -7, (D@@3)(y)(0) = 15],y(t), singsol=all)
\[ y \left (t \right ) = {\mathrm e}^{-t}-2 \,{\mathrm e}^{-2 t} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 16
DSolve[{y''''[t]-5*y''[t]+4*y[t]==0,{y[0]==-1,y'[0]==3,y''[0]==-7,y'''[0]==15}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to e^{-2 t} \left (e^t-2\right ) \]