Internal problem ID [2133]
Internal file name [OUTPUT/2133_Monday_February_26_2024_09_17_47_AM_96573020/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 18, page 82
Problem number: 17.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }-6 y^{\prime }+2 y=0} \] The characteristic equation is \[ \lambda ^{4}-2 \lambda ^{3}-6 \lambda +2 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =1\right )\\ \lambda _2 &= \operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =2\right )\\ \lambda _3 &= \operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =3\right )\\ \lambda _4 &= \operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =4\right ) \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =2\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =3\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =4\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =1\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =2\right ) x}\\ y_2 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =3\right ) x}\\ y_3 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =4\right ) x}\\ y_4 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =1\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} &=2\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} &=3\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} &=4\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} &=1\right ) x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =2\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =3\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =4\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =1\right ) x} c_{4} \] Verified OK.
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: 32
dsolve(diff(y(x),x$4)-2*diff(y(x),x$3)-6*diff(y(x),x)+2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-6 \textit {\_Z} +2, \operatorname {index} =\textit {\_a} \right ) x} \textit {\_C}_{\textit {\_a}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 114
DSolve[y''''[x]-2*y'''[x]-6*y'[x]+2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^3-6 \text {$\#$1}+2\&,1\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^3-6 \text {$\#$1}+2\&,3\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^3-6 \text {$\#$1}+2\&,4\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^4-2 \text {$\#$1}^3-6 \text {$\#$1}+2\&,2\right ]\right ) \]