Internal problem ID [13631]
Internal file name [OUTPUT/12803_Saturday_February_17_2024_08_44_53_AM_9370537/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 19. Arbitrary Homogeneous linear equations with constant coefficients.
Additional exercises page 369
Problem number: 19.3 (d).
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]]
\[ \boxed {y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }+9 y^{\prime \prime }+9 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 10, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 6, y^{\prime \prime \prime }\left (0\right ) = -60] \end {align*}
The characteristic equation is \[ \lambda ^{4}+\lambda ^{3}+9 \lambda ^{2}+9 \lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= -1\\ \lambda _3 &= 3 i\\ \lambda _4 &= -3 i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-x} c_{1} +c_{2} +{\mathrm e}^{-3 i x} c_{3} +{\mathrm e}^{3 i x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-x}\\ y_2 &= 1\\ y_3 &= {\mathrm e}^{-3 i x}\\ y_4 &= {\mathrm e}^{3 i x} \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = {\mathrm e}^{-x} c_{1} +c_{2} +{\mathrm e}^{-3 i x} c_{3} +{\mathrm e}^{3 i x} c_{4} \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 \(y = 10\) and \(x = 0\) in the above gives \begin {align*} 10 = c_{1} +c_{2} +c_{3} +c_{4}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = -{\mathrm e}^{-x} c_{1} -3 i {\mathrm e}^{-3 i x} c_{3} +3 i {\mathrm e}^{3 i x} c_{4} \end {align*}
substituting \(y^{\prime } = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = -3 c_{3} i+3 c_{4} i-c_{1}\tag {2A} \end {align*}
Taking two derivatives of the solution gives \begin {align*} y^{\prime \prime } = {\mathrm e}^{-x} c_{1} -9 \,{\mathrm e}^{-3 i x} c_{3} -9 \,{\mathrm e}^{3 i x} c_{4} \end {align*}
substituting \(y^{\prime \prime } = 6\) and \(x = 0\) in the above gives \begin {align*} 6 = c_{1} -9 c_{3} -9 c_{4}\tag {3A} \end {align*}
Taking three derivatives of the solution gives \begin {align*} y^{\prime \prime \prime } = -{\mathrm e}^{-x} c_{1} +27 i {\mathrm e}^{-3 i x} c_{3} -27 i {\mathrm e}^{3 i x} c_{4} \end {align*}
substituting \(y^{\prime \prime \prime } = -60\) and \(x = 0\) in the above gives \begin {align*} -60 = 27 c_{3} i-27 c_{4} i-c_{1}\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}&=6\\ c_{2}&=4\\ c_{3}&=i\\ c_{4}&=-i \end {align*}
Substituting these values back in above solution results in \begin {align*} y = 6 \,{\mathrm e}^{-x}+4+2 \sin \left (3 x \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 6 \,{\mathrm e}^{-x}+4+2 \sin \left (3 x \right ) \\ \end{align*}
Verification of solutions
\[ y = 6 \,{\mathrm e}^{-x}+4+2 \sin \left (3 x \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }+9 y^{\prime \prime }+9 y^{\prime }=0, y \left (0\right )=10, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=0, y^{\prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=6, y^{\prime \prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=-60\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 (x \right ) \\ {} & {} & y_{1}\left (x \right )=y \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (x \right ) \\ {} & {} & y_{2}\left (x \right )=y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (x \right ) \\ {} & {} & y_{3}\left (x \right )=y^{\prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{4}\left (x \right ) \\ {} & {} & y_{4}\left (x \right )=y^{\prime \prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{4}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{4}^{\prime }\left (x \right )=-y_{4}\left (x \right )-9 y_{3}\left (x \right )-9 y_{2}\left (x \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (x \right )=y_{1}^{\prime }\left (x \right ), y_{3}\left (x \right )=y_{2}^{\prime }\left (x \right ), y_{4}\left (x \right )=y_{3}^{\prime }\left (x \right ), y_{4}^{\prime }\left (x \right )=-y_{4}\left (x \right )-9 y_{3}\left (x \right )-9 y_{2}\left (x \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=\left [\begin {array}{c} y_{1}\left (x \right ) \\ y_{2}\left (x \right ) \\ y_{3}\left (x \right ) \\ y_{4}\left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & -9 & -9 & -1 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \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 \\ 0 & -9 & -9 & -1 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=A \cdot {\moverset {\rightarrow }{y}}\left (x \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 [-1, \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ]\right ], \left [0, \left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ], \left [-3 \,\mathrm {I}, \left [\begin {array}{c} -\frac {\mathrm {I}}{27} \\ -\frac {1}{9} \\ \frac {\mathrm {I}}{3} \\ 1 \end {array}\right ]\right ], \left [3 \,\mathrm {I}, \left [\begin {array}{c} \frac {\mathrm {I}}{27} \\ -\frac {1}{9} \\ -\frac {\mathrm {I}}{3} \\ 1 \end {array}\right ]\right ]\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}}_{1}={\mathrm e}^{-x}\cdot \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [0, \left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}=\left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-3 \,\mathrm {I}, \left [\begin {array}{c} -\frac {\mathrm {I}}{27} \\ -\frac {1}{9} \\ \frac {\mathrm {I}}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-3 \,\mathrm {I} x}\cdot \left [\begin {array}{c} -\frac {\mathrm {I}}{27} \\ -\frac {1}{9} \\ \frac {\mathrm {I}}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Use Euler identity to write solution in terms of}\hspace {3pt} \sin \hspace {3pt}\textrm {and}\hspace {3pt} \cos \\ {} & {} & \left (\cos \left (3 x \right )-\mathrm {I} \sin \left (3 x \right )\right )\cdot \left [\begin {array}{c} -\frac {\mathrm {I}}{27} \\ -\frac {1}{9} \\ \frac {\mathrm {I}}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} -\frac {\mathrm {I}}{27} \left (\cos \left (3 x \right )-\mathrm {I} \sin \left (3 x \right )\right ) \\ -\frac {\cos \left (3 x \right )}{9}+\frac {\mathrm {I} \sin \left (3 x \right )}{9} \\ \frac {\mathrm {I}}{3} \left (\cos \left (3 x \right )-\mathrm {I} \sin \left (3 x \right )\right ) \\ \cos \left (3 x \right )-\mathrm {I} \sin \left (3 x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{3}\left (x \right )=\left [\begin {array}{c} -\frac {\sin \left (3 x \right )}{27} \\ -\frac {\cos \left (3 x \right )}{9} \\ \frac {\sin \left (3 x \right )}{3} \\ \cos \left (3 x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )=\left [\begin {array}{c} -\frac {\cos \left (3 x \right )}{27} \\ \frac {\sin \left (3 x \right )}{9} \\ \frac {\cos \left (3 x \right )}{3} \\ -\sin \left (3 x \right ) \end {array}\right ]\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}\left (x \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (x \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}={\mathrm e}^{-x} c_{1} \cdot \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ]+\left [\begin {array}{c} c_{2} -\frac {c_{3} \sin \left (3 x \right )}{27}-\frac {c_{4} \cos \left (3 x \right )}{27} \\ -\frac {c_{3} \cos \left (3 x \right )}{9}+\frac {c_{4} \sin \left (3 x \right )}{9} \\ \frac {c_{3} \sin \left (3 x \right )}{3}+\frac {c_{4} \cos \left (3 x \right )}{3} \\ c_{3} \cos \left (3 x \right )-c_{4} \sin \left (3 x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=-{\mathrm e}^{-x} c_{1} -\frac {c_{4} \cos \left (3 x \right )}{27}-\frac {c_{3} \sin \left (3 x \right )}{27}+c_{2} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y \left (0\right )=10 \\ {} & {} & 10=-c_{1} -\frac {c_{4}}{27}+c_{2} \\ \bullet & {} & \textrm {Calculate the 1st derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }={\mathrm e}^{-x} c_{1} +\frac {c_{4} \sin \left (3 x \right )}{9}-\frac {c_{3} \cos \left (3 x \right )}{9} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=c_{1} -\frac {c_{3}}{9} \\ \bullet & {} & \textrm {Calculate the 2nd derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=-{\mathrm e}^{-x} c_{1} +\frac {c_{4} \cos \left (3 x \right )}{3}+\frac {c_{3} \sin \left (3 x \right )}{3} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=6 \\ {} & {} & 6=-c_{1} +\frac {c_{4}}{3} \\ \bullet & {} & \textrm {Calculate the 3rd derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }={\mathrm e}^{-x} c_{1} -c_{4} \sin \left (3 x \right )+c_{3} \cos \left (3 x \right ) \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime \prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=-60 \\ {} & {} & -60=c_{1} +c_{3} \\ \bullet & {} & \textrm {Solve for the unknown coefficients}\hspace {3pt} \\ {} & {} & \left \{c_{1} =-6, c_{2} =4, c_{3} =-54, c_{4} =0\right \} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=6 \,{\mathrm e}^{-x}+4+2 \sin \left (3 x \right ) \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: 18
dsolve([diff(y(x),x$4)+diff(y(x),x$3)+9*diff(y(x),x$2)+9*diff(y(x),x)=0,y(0) = 10, D(y)(0) = 0, (D@@2)(y)(0) = 6, (D@@3)(y)(0) = -60],y(x), singsol=all)
\[ y \left (x \right ) = 4+6 \,{\mathrm e}^{-x}+2 \sin \left (3 x \right ) \]
✓ Solution by Mathematica
Time used: 0.055 (sec). Leaf size: 20
DSolve[{y''''[x]+y'''[x]+9*y''[x]+9*y'[x]==0,{y[0]==10,y'[0]==0,y''[0]==6,y'''[0]==-60}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 6 e^{-x}+2 \sin (3 x)+4 \]