Internal problem ID [1491]
Internal file name [OUTPUT/1492_Sunday_June_05_2022_02_19_34_AM_95161317/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 9 Introduction to Linear Higher Order Equations. Section 9.2. constant
coefficient. Page 483
Problem number: section 9.2, problem 27.
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]]
\[ \boxed {4 y^{\prime \prime \prime \prime }+8 y^{\prime \prime \prime }+19 y^{\prime \prime }+32 y^{\prime }+12 y=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -3, y^{\prime \prime }\left (0\right ) = -{\frac {7}{2}}, y^{\prime \prime \prime }\left (0\right ) = {\frac {31}{4}}\right ] \end {align*}
The characteristic equation is \[ 4 \lambda ^{4}+8 \lambda ^{3}+19 \lambda ^{2}+32 \lambda +12 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -{\frac {3}{2}}\\ \lambda _2 &= -{\frac {1}{2}}\\ \lambda _3 &= 2 i\\ \lambda _4 &= -2 i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-\frac {3 x}{2}}+c_{2} {\mathrm e}^{-\frac {x}{2}}+{\mathrm e}^{-2 i x} c_{3} +{\mathrm e}^{2 i x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-\frac {3 x}{2}}\\ y_2 &= {\mathrm e}^{-\frac {x}{2}}\\ y_3 &= {\mathrm e}^{-2 i x}\\ y_4 &= {\mathrm e}^{2 i x} \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}^{-\frac {3 x}{2}}+c_{2} {\mathrm e}^{-\frac {x}{2}}+{\mathrm e}^{-2 i x} c_{3} +{\mathrm e}^{2 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 \(y = 3\) and \(x = 0\) in the above gives \begin {align*} 3 = c_{1} +c_{2} +c_{3} +c_{4}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = -\frac {3 c_{1} {\mathrm e}^{-\frac {3 x}{2}}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {x}{2}}}{2}-2 i {\mathrm e}^{-2 i x} c_{3} +2 i {\mathrm e}^{2 i x} c_{4} \end {align*}
substituting \(y^{\prime } = -3\) and \(x = 0\) in the above gives \begin {align*} -3 = -\frac {3}{2} c_{1} -\frac {1}{2} c_{2} -2 i c_{3} +2 i c_{4}\tag {2A} \end {align*}
Taking two derivatives of the solution gives \begin {align*} y^{\prime \prime } = \frac {9 c_{1} {\mathrm e}^{-\frac {3 x}{2}}}{4}+\frac {c_{2} {\mathrm e}^{-\frac {x}{2}}}{4}-4 \,{\mathrm e}^{-2 i x} c_{3} -4 \,{\mathrm e}^{2 i x} c_{4} \end {align*}
substituting \(y^{\prime \prime } = -{\frac {7}{2}}\) and \(x = 0\) in the above gives \begin {align*} -{\frac {7}{2}} = \frac {9 c_{1}}{4}+\frac {c_{2}}{4}-4 c_{3} -4 c_{4}\tag {3A} \end {align*}
Taking three derivatives of the solution gives \begin {align*} y^{\prime \prime \prime } = -\frac {27 c_{1} {\mathrm e}^{-\frac {3 x}{2}}}{8}-\frac {c_{2} {\mathrm e}^{-\frac {x}{2}}}{8}+8 i {\mathrm e}^{-2 i x} c_{3} -8 i {\mathrm e}^{2 i x} c_{4} \end {align*}
substituting \(y^{\prime \prime \prime } = {\frac {31}{4}}\) and \(x = 0\) in the above gives \begin {align*} {\frac {31}{4}} = -\frac {27}{8} c_{1} -\frac {1}{8} c_{2} +8 i c_{3} -8 i 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}&=0\\ c_{2}&=2\\ c_{3}&=\frac {1}{2}-\frac {i}{2}\\ c_{4}&=\frac {1}{2}+\frac {i}{2} \end {align*}
Substituting these values back in above solution results in \begin {align*} y = 2 \,{\mathrm e}^{-\frac {x}{2}}+\cos \left (2 x \right )-\sin \left (2 x \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \,{\mathrm e}^{-\frac {x}{2}}+\cos \left (2 x \right )-\sin \left (2 x \right ) \\ \end{align*}
Verification of solutions
\[ y = 2 \,{\mathrm e}^{-\frac {x}{2}}+\cos \left (2 x \right )-\sin \left (2 x \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [4 y^{\prime \prime \prime \prime }+8 y^{\prime \prime \prime }+19 y^{\prime \prime }+32 y^{\prime }+12 y=0, y \left (0\right )=3, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=-3, y^{\prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=-\frac {7}{2}, y^{\prime \prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=\frac {31}{4}\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & y^{\prime \prime \prime \prime } \\ \bullet & {} & \textrm {Isolate 4th derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }=-2 y^{\prime \prime \prime }-\frac {19 y^{\prime \prime }}{4}-8 y^{\prime }-3 y \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+\frac {19 y^{\prime \prime }}{4}+8 y^{\prime }+3 y=0 \\ \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 )=-2 y_{4}\left (x \right )-\frac {19 y_{3}\left (x \right )}{4}-8 y_{2}\left (x \right )-3 y_{1}\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 )=-2 y_{4}\left (x \right )-\frac {19 y_{3}\left (x \right )}{4}-8 y_{2}\left (x \right )-3 y_{1}\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 \\ -3 & -8 & -\frac {19}{4} & -2 \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 \\ -3 & -8 & -\frac {19}{4} & -2 \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 [-\frac {3}{2}, \left [\begin {array}{c} -\frac {8}{27} \\ \frac {4}{9} \\ -\frac {2}{3} \\ 1 \end {array}\right ]\right ], \left [-\frac {1}{2}, \left [\begin {array}{c} -8 \\ 4 \\ -2 \\ 1 \end {array}\right ]\right ], \left [-2 \,\mathrm {I}, \left [\begin {array}{c} -\frac {\mathrm {I}}{8} \\ -\frac {1}{4} \\ \frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ], \left [2 \,\mathrm {I}, \left [\begin {array}{c} \frac {\mathrm {I}}{8} \\ -\frac {1}{4} \\ -\frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {3}{2}, \left [\begin {array}{c} -\frac {8}{27} \\ \frac {4}{9} \\ -\frac {2}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{-\frac {3 x}{2}}\cdot \left [\begin {array}{c} -\frac {8}{27} \\ \frac {4}{9} \\ -\frac {2}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {1}{2}, \left [\begin {array}{c} -8 \\ 4 \\ -2 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}={\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} -8 \\ 4 \\ -2 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-2 \,\mathrm {I}, \left [\begin {array}{c} -\frac {\mathrm {I}}{8} \\ -\frac {1}{4} \\ \frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-2 \,\mathrm {I} x}\cdot \left [\begin {array}{c} -\frac {\mathrm {I}}{8} \\ -\frac {1}{4} \\ \frac {\mathrm {I}}{2} \\ 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 (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right )\cdot \left [\begin {array}{c} -\frac {\mathrm {I}}{8} \\ -\frac {1}{4} \\ \frac {\mathrm {I}}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} -\frac {\mathrm {I}}{8} \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right ) \\ -\frac {\cos \left (2 x \right )}{4}+\frac {\mathrm {I} \sin \left (2 x \right )}{4} \\ \frac {\mathrm {I}}{2} \left (\cos \left (2 x \right )-\mathrm {I} \sin \left (2 x \right )\right ) \\ \cos \left (2 x \right )-\mathrm {I} \sin \left (2 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 (2 x \right )}{8} \\ -\frac {\cos \left (2 x \right )}{4} \\ \frac {\sin \left (2 x \right )}{2} \\ \cos \left (2 x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )=\left [\begin {array}{c} -\frac {\cos \left (2 x \right )}{8} \\ \frac {\sin \left (2 x \right )}{4} \\ \frac {\cos \left (2 x \right )}{2} \\ -\sin \left (2 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}}=c_{1} {\mathrm e}^{-\frac {3 x}{2}}\cdot \left [\begin {array}{c} -\frac {8}{27} \\ \frac {4}{9} \\ -\frac {2}{3} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} -8 \\ 4 \\ -2 \\ 1 \end {array}\right ]+\left [\begin {array}{c} -\frac {c_{3} \sin \left (2 x \right )}{8}-\frac {c_{4} \cos \left (2 x \right )}{8} \\ -\frac {c_{3} \cos \left (2 x \right )}{4}+\frac {c_{4} \sin \left (2 x \right )}{4} \\ \frac {c_{3} \sin \left (2 x \right )}{2}+\frac {c_{4} \cos \left (2 x \right )}{2} \\ c_{3} \cos \left (2 x \right )-c_{4} \sin \left (2 x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=-\frac {8 c_{1} {\mathrm e}^{-\frac {3 x}{2}}}{27}-8 c_{2} {\mathrm e}^{-\frac {x}{2}}-\frac {c_{4} \cos \left (2 x \right )}{8}-\frac {c_{3} \sin \left (2 x \right )}{8} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y \left (0\right )=3 \\ {} & {} & 3=-\frac {8 c_{1}}{27}-8 c_{2} -\frac {c_{4}}{8} \\ \bullet & {} & \textrm {Calculate the 1st derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {4 c_{1} {\mathrm e}^{-\frac {3 x}{2}}}{9}+4 c_{2} {\mathrm e}^{-\frac {x}{2}}+\frac {c_{4} \sin \left (2 x \right )}{4}-\frac {c_{3} \cos \left (2 x \right )}{4} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=-3 \\ {} & {} & -3=\frac {4 c_{1}}{9}+4 c_{2} -\frac {c_{3}}{4} \\ \bullet & {} & \textrm {Calculate the 2nd derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=-\frac {2 c_{1} {\mathrm e}^{-\frac {3 x}{2}}}{3}-2 c_{2} {\mathrm e}^{-\frac {x}{2}}+\frac {c_{4} \cos \left (2 x \right )}{2}+\frac {c_{3} \sin \left (2 x \right )}{2} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=-\frac {7}{2} \\ {} & {} & -\frac {7}{2}=-\frac {2 c_{1}}{3}-2 c_{2} +\frac {c_{4}}{2} \\ \bullet & {} & \textrm {Calculate the 3rd derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }=c_{1} {\mathrm e}^{-\frac {3 x}{2}}+c_{2} {\mathrm e}^{-\frac {x}{2}}-c_{4} \sin \left (2 x \right )+c_{3} \cos \left (2 x \right ) \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime \prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=\frac {31}{4} \\ {} & {} & \frac {31}{4}=c_{1} +c_{2} +c_{3} \\ \bullet & {} & \textrm {Solve for the unknown coefficients}\hspace {3pt} \\ {} & {} & \left \{c_{1} =0, c_{2} =-\frac {1}{4}, c_{3} =8, c_{4} =-8\right \} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=2 \,{\mathrm e}^{-\frac {x}{2}}+\cos \left (2 x \right )-\sin \left (2 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: 0.031 (sec). Leaf size: 21
dsolve([4*diff(y(x),x$4)+8*diff(y(x),x$3)+19*diff(y(x),x$2)+32*diff(y(x),x)+12*y(x)=0,y(0) = 3, D(y)(0) = -3, (D@@2)(y)(0) = -7/2, (D@@3)(y)(0) = 31/4],y(x), singsol=all)
\[ y \left (x \right ) = 2 \,{\mathrm e}^{-\frac {x}{2}}-\sin \left (2 x \right )+\cos \left (2 x \right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 31
DSolve[{y''''[x]+2*y'''[x]-2*y''[x]-8*y'[x]-8*y[x]==0,{y[0]==5,y'[0]==-2,y''[0]==6,y'''[0]==8}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-2 x} \left (e^{4 x}+e^x \sin (x)+3 e^x \cos (x)+1\right ) \]