Internal problem ID [1489]
Internal file name [OUTPUT/1490_Sunday_June_05_2022_02_19_30_AM_21780042/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 25.
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]]
\[ \boxed {4 y^{\prime \prime \prime \prime }-13 y^{\prime \prime }+9 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 3, y^{\prime \prime }\left (0\right ) = 1, y^{\prime \prime \prime }\left (0\right ) = 3] \end {align*}
The characteristic equation is \[ 4 \lambda ^{4}-13 \lambda ^{2}+9 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1\\ \lambda _2 &= -1\\ \lambda _3 &= -{\frac {3}{2}}\\ \lambda _4 &= {\frac {3}{2}} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{-\frac {3 x}{2}} c_{3} +{\mathrm e}^{\frac {3 x}{2}} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-x}\\ y_2 &= {\mathrm e}^{x}\\ y_3 &= {\mathrm e}^{-\frac {3 x}{2}}\\ y_4 &= {\mathrm e}^{\frac {3 x}{2}} \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}^{-x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{-\frac {3 x}{2}} c_{3} +{\mathrm e}^{\frac {3 x}{2}} 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 \(y = 1\) and \(x = 0\) 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}^{-x}+c_{2} {\mathrm e}^{x}-\frac {3 \,{\mathrm e}^{-\frac {3 x}{2}} c_{3}}{2}+\frac {3 \,{\mathrm e}^{\frac {3 x}{2}} c_{4}}{2} \end {align*}
substituting \(y^{\prime } = 3\) and \(x = 0\) in the above gives \begin {align*} 3 = -c_{1} +c_{2} -\frac {3 c_{3}}{2}+\frac {3 c_{4}}{2}\tag {2A} \end {align*}
Taking two derivatives of the solution gives \begin {align*} y^{\prime \prime } = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}+\frac {9 \,{\mathrm e}^{-\frac {3 x}{2}} c_{3}}{4}+\frac {9 \,{\mathrm e}^{\frac {3 x}{2}} c_{4}}{4} \end {align*}
substituting \(y^{\prime \prime } = 1\) and \(x = 0\) in the above gives \begin {align*} 1 = c_{1} +c_{2} +\frac {9 c_{3}}{4}+\frac {9 c_{4}}{4}\tag {3A} \end {align*}
Taking three derivatives of the solution gives \begin {align*} y^{\prime \prime \prime } = -c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}-\frac {27 \,{\mathrm e}^{-\frac {3 x}{2}} c_{3}}{8}+\frac {27 \,{\mathrm e}^{\frac {3 x}{2}} c_{4}}{8} \end {align*}
substituting \(y^{\prime \prime \prime } = 3\) and \(x = 0\) in the above gives \begin {align*} 3 = -c_{1} +c_{2} -\frac {27 c_{3}}{8}+\frac {27 c_{4}}{8}\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}^{-x}+2 \,{\mathrm e}^{x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -{\mathrm e}^{-x}+2 \,{\mathrm e}^{x} \\ \end{align*}
Verification of solutions
\[ y = -{\mathrm e}^{-x}+2 \,{\mathrm e}^{x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [4 y^{\prime \prime \prime \prime }-13 y^{\prime \prime }+9 y=0, y \left (0\right )=1, 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 \}}}}=1, y^{\prime \prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=3\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 }=\frac {13 y^{\prime \prime }}{4}-\frac {9 y}{4} \\ \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 }-\frac {13 y^{\prime \prime }}{4}+\frac {9 y}{4}=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 )=\frac {13 y_{3}\left (x \right )}{4}-\frac {9 y_{1}\left (x \right )}{4} \\ & {} & \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 )=\frac {13 y_{3}\left (x \right )}{4}-\frac {9 y_{1}\left (x \right )}{4}\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 \\ -\frac {9}{4} & 0 & \frac {13}{4} & 0 \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 \\ -\frac {9}{4} & 0 & \frac {13}{4} & 0 \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 [-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 [\frac {3}{2}, \left [\begin {array}{c} \frac {8}{27} \\ \frac {4}{9} \\ \frac {2}{3} \\ 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 [-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}^{-x}\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}^{x}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\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}}_{4}={\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 {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}^{-\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}^{-x}\cdot \left [\begin {array}{c} -1 \\ 1 \\ -1 \\ 1 \end {array}\right ]+{\mathrm e}^{x} c_{3} \cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ]+{\mathrm e}^{\frac {3 x}{2}} c_{4} \cdot \left [\begin {array}{c} \frac {8}{27} \\ \frac {4}{9} \\ \frac {2}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (8 \,{\mathrm e}^{3 x} c_{4} +27 \,{\mathrm e}^{\frac {5 x}{2}} c_{3} -27 c_{2} {\mathrm e}^{\frac {x}{2}}-8 c_{1} \right ) {\mathrm e}^{-\frac {3 x}{2}}}{27} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y \left (0\right )=1 \\ {} & {} & 1=\frac {8 c_{4}}{27}+c_{3} -c_{2} -\frac {8 c_{1}}{27} \\ \bullet & {} & \textrm {Calculate the 1st derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (24 \,{\mathrm e}^{3 x} c_{4} +\frac {135 \,{\mathrm e}^{\frac {5 x}{2}} c_{3}}{2}-\frac {27 c_{2} {\mathrm e}^{\frac {x}{2}}}{2}\right ) {\mathrm e}^{-\frac {3 x}{2}}}{27}-\frac {\left (8 \,{\mathrm e}^{3 x} c_{4} +27 \,{\mathrm e}^{\frac {5 x}{2}} c_{3} -27 c_{2} {\mathrm e}^{\frac {x}{2}}-8 c_{1} \right ) {\mathrm e}^{-\frac {3 x}{2}}}{18} \\ \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_{4}}{9}+c_{3} +c_{2} +\frac {4 c_{1}}{9} \\ \bullet & {} & \textrm {Calculate the 2nd derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=\frac {\left (72 \,{\mathrm e}^{3 x} c_{4} +\frac {675 \,{\mathrm e}^{\frac {5 x}{2}} c_{3}}{4}-\frac {27 c_{2} {\mathrm e}^{\frac {x}{2}}}{4}\right ) {\mathrm e}^{-\frac {3 x}{2}}}{27}-\frac {\left (24 \,{\mathrm e}^{3 x} c_{4} +\frac {135 \,{\mathrm e}^{\frac {5 x}{2}} c_{3}}{2}-\frac {27 c_{2} {\mathrm e}^{\frac {x}{2}}}{2}\right ) {\mathrm e}^{-\frac {3 x}{2}}}{9}+\frac {\left (8 \,{\mathrm e}^{3 x} c_{4} +27 \,{\mathrm e}^{\frac {5 x}{2}} c_{3} -27 c_{2} {\mathrm e}^{\frac {x}{2}}-8 c_{1} \right ) {\mathrm e}^{-\frac {3 x}{2}}}{12} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=1 \\ {} & {} & 1=\frac {2 c_{4}}{3}+c_{3} -c_{2} -\frac {2 c_{1}}{3} \\ \bullet & {} & \textrm {Calculate the 3rd derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }=\frac {\left (216 \,{\mathrm e}^{3 x} c_{4} +\frac {3375 \,{\mathrm e}^{\frac {5 x}{2}} c_{3}}{8}-\frac {27 c_{2} {\mathrm e}^{\frac {x}{2}}}{8}\right ) {\mathrm e}^{-\frac {3 x}{2}}}{27}-\frac {\left (72 \,{\mathrm e}^{3 x} c_{4} +\frac {675 \,{\mathrm e}^{\frac {5 x}{2}} c_{3}}{4}-\frac {27 c_{2} {\mathrm e}^{\frac {x}{2}}}{4}\right ) {\mathrm e}^{-\frac {3 x}{2}}}{6}+\frac {\left (24 \,{\mathrm e}^{3 x} c_{4} +\frac {135 \,{\mathrm e}^{\frac {5 x}{2}} c_{3}}{2}-\frac {27 c_{2} {\mathrm e}^{\frac {x}{2}}}{2}\right ) {\mathrm e}^{-\frac {3 x}{2}}}{4}-\frac {\left (8 \,{\mathrm e}^{3 x} c_{4} +27 \,{\mathrm e}^{\frac {5 x}{2}} c_{3} -27 c_{2} {\mathrm e}^{\frac {x}{2}}-8 c_{1} \right ) {\mathrm e}^{-\frac {3 x}{2}}}{8} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime \prime \prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=3 \\ {} & {} & 3=c_{1} +c_{2} +c_{3} +c_{4} \\ \bullet & {} & \textrm {Solve for the unknown coefficients}\hspace {3pt} \\ {} & {} & \left \{c_{1} =0, c_{2} =1, c_{3} =2, c_{4} =0\right \} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-{\mathrm e}^{-x}+2 \,{\mathrm e}^{x} \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: 15
dsolve([4*diff(y(x),x$4)-13*diff(y(x),x$2)+9*y(x)=0,y(0) = 1, D(y)(0) = 3, (D@@2)(y)(0) = 1, (D@@3)(y)(0) = 3],y(x), singsol=all)
\[ y \left (x \right ) = -{\mathrm e}^{-x}+2 \,{\mathrm e}^{x} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 18
DSolve[{4*y''''[x]-13*y''[x]+9*y[x]==0,{y[0]==1,y'[0]==3,y''[0]==1,y'''[0]==3}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2 e^x-e^{-x} \]