14.27 problem 27

Internal problem ID [2228]
Internal file name [OUTPUT/2228_Monday_February_26_2024_09_18_33_AM_59470393/index.tex]

Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section: Exercise 23, page 106
Problem number: 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, _linear, _nonhomogeneous]]

\[ \boxed {y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-y^{\prime }+2 y=\sin \left (2 x \right )} \] This is higher order nonhomogeneous ODE. Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE And \(y_p\) is a particular solution to the nonhomogeneous ODE. \(y_h\) is the solution to \[ y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-y^{\prime }+2 y = 0 \] The characteristic equation is \[ \lambda ^{4}+3 \lambda ^{2}-\lambda +2 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =1\right )\\ \lambda _2 &= \operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =2\right )\\ \lambda _3 &= \operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =3\right )\\ \lambda _4 &= \operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =4\right ) \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =3\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =1\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =2\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =4\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=3\right ) x} \\ y_2 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=1\right ) x} \\ y_3 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=2\right ) x} \\ y_4 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=4\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-y^{\prime }+2 y = \sin \left (2 x \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ \sin \left (2 x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{\cos \left (2 x \right ), \sin \left (2 x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =1\right ) x}, {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =2\right ) x}, {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =3\right ) x}, {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =4\right ) x}\} \] Since there is no duplication between the basis function in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis in the UC_set. \[ y_p = A_{1} \cos \left (2 x \right )+A_{2} \sin \left (2 x \right ) \] The unknowns \(\{A_{1}, A_{2}\}\) are found by substituting the above trial solution \(y_p\) into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives \[ 6 A_{1} \cos \left (2 x \right )+6 A_{2} \sin \left (2 x \right )+2 A_{1} \sin \left (2 x \right )-2 A_{2} \cos \left (2 x \right ) = \sin \left (2 x \right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = {\frac {1}{20}}, A_{2} = {\frac {3}{20}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {\cos \left (2 x \right )}{20}+\frac {3 \sin \left (2 x \right )}{20} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=3\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=1\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=2\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=4\right ) x} c_{4}\right ) + \left (\frac {\cos \left (2 x \right )}{20}+\frac {3 \sin \left (2 x \right )}{20}\right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=3\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=1\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=2\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} &=4\right ) x} c_{4} +\frac {\cos \left (2 x \right )}{20}+\frac {3 \sin \left (2 x \right )}{20} \\ \end{align*}

Verification of solutions

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =3\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =1\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =2\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-\textit {\_Z} +2, \operatorname {index} =4\right ) x} c_{4} +\frac {\cos \left (2 x \right )}{20}+\frac {3 \sin \left (2 x \right )}{20} \] Verified OK.

Maple trace

`Methods for high order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 4; linear nonhomogeneous with symmetry [0,1] 
trying high order linear exact nonhomogeneous 
trying differential order: 4; missing the dependent variable 
checking if the LODE has constant coefficients 
<- constant coefficients successful`
 

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 6182

dsolve(diff(y(x),x$4)+3*diff(y(x),x$2)-diff(y(x),x)+2*y(x)=sin(2*x),y(x), singsol=all)
 

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 1.487 (sec). Leaf size: 1124

DSolve[y''''[x]+3*y''[x]-y'[x]+2*y[x]==Sin[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{x \text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,1\right ]} c_1+e^{x \text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]} c_2+e^{x \text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]} c_3+e^{x \text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]} c_4-\frac {\left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,1\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,1\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]\right ) \left (2 \cos (2 x)+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ] \sin (2 x)\right )}{\sqrt {761} \left (4+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]^2\right )}+\frac {\left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,1\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,1\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]\right ) \left (2 \cos (2 x)+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ] \sin (2 x)\right )}{\sqrt {761} \left (4+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]^2\right )}-\frac {\left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,1\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,1\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]\right ) \left (2 \cos (2 x)+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ] \sin (2 x)\right )}{\sqrt {761} \left (4+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]^2\right )}-\frac {e^{\left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]\right ) x+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,1\right ] x} \left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]-\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]\right ) \left (\left (\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]\right ) \sin (2 x)-2 \cos (2 x)\right )}{\sqrt {761} \left (-2 i+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]\right ) \left (2 i+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,2\right ]+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,3\right ]+\text {Root}\left [\text {$\#$1}^4+3 \text {$\#$1}^2-\text {$\#$1}+2\&,4\right ]\right )} \]