Internal problem ID [12779]
Internal file name [OUTPUT/11432_Friday_November_03_2023_06_32_53_AM_50427656/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 4. N-th Order Linear Differential Equations. Exercises 4.4, page 218
Problem number: 7.
ODE order: 6.
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]]
\[ \boxed {y^{\left (6\right )}-12 y^{\left (5\right )}+63 y^{\prime \prime \prime \prime }-18 y^{\prime \prime \prime }+315 y^{\prime \prime }-300 y^{\prime }+125 y={\mathrm e}^{x} \left (48 \cos \left (x \right )+96 \sin \left (x \right )\right )} \] 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 \[ y^{\left (6\right )}-12 y^{\left (5\right )}+63 y^{\prime \prime \prime \prime }-18 y^{\prime \prime \prime }+315 y^{\prime \prime }-300 y^{\prime }+125 y = 0 \] The characteristic equation is \[ \lambda ^{6}-12 \lambda ^{5}+63 \lambda ^{4}-18 \lambda ^{3}+315 \lambda ^{2}-300 \lambda +125 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =1\right )\\ \lambda _2 &= \operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =2\right )\\ \lambda _3 &= \operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =3\right )\\ \lambda _4 &= \operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =4\right )\\ \lambda _5 &= \operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =5\right )\\ \lambda _6 &= \operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =6\right ) \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =2\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =5\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =6\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =3\right ) x} c_{4} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =1\right ) x} c_{5} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =4\right ) x} c_{6} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=2\right ) x} \\ y_2 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=5\right ) x} \\ y_3 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=6\right ) x} \\ y_4 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=3\right ) x} \\ y_5 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=1\right ) x} \\ y_6 &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=4\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\left (6\right )}-12 y^{\left (5\right )}+63 y^{\prime \prime \prime \prime }-18 y^{\prime \prime \prime }+315 y^{\prime \prime }-300 y^{\prime }+125 y = {\mathrm e}^{x} \left (48 \cos \left (x \right )+96 \sin \left (x \right )\right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ {\mathrm e}^{x} \left (48 \cos \left (x \right )+96 \sin \left (x \right )\right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{x} \cos \left (x \right ), {\mathrm e}^{x} \sin \left (x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =1\right ) x}, {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =2\right ) x}, {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =3\right ) x}, {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =4\right ) x}, {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =5\right ) x}, {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =6\right ) x}\} \] 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. \[ y_p = A_{1} {\mathrm e}^{x} \cos \left (x \right )+A_{2} {\mathrm e}^{x} \sin \left (x \right ) \] 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 \[ -334 A_{1} {\mathrm e}^{x} \sin \left (x \right )+334 A_{2} {\mathrm e}^{x} \cos \left (x \right )-343 A_{1} {\mathrm e}^{x} \cos \left (x \right )-343 A_{2} {\mathrm e}^{x} \sin \left (x \right ) = {\mathrm e}^{x} \left (48 \cos \left (x \right )+96 \sin \left (x \right )\right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = -{\frac {48528}{229205}}, A_{2} = -{\frac {16896}{229205}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -\frac {48528 \,{\mathrm e}^{x} \cos \left (x \right )}{229205}-\frac {16896 \,{\mathrm e}^{x} \sin \left (x \right )}{229205} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=2\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=5\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=6\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=3\right ) x} c_{4} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=1\right ) x} c_{5} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=4\right ) x} c_{6}\right ) + \left (-\frac {48528 \,{\mathrm e}^{x} \cos \left (x \right )}{229205}-\frac {16896 \,{\mathrm e}^{x} \sin \left (x \right )}{229205}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=2\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=5\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=6\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=3\right ) x} c_{4} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=1\right ) x} c_{5} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} &=4\right ) x} c_{6} -\frac {48528 \,{\mathrm e}^{x} \cos \left (x \right )}{229205}-\frac {16896 \,{\mathrm e}^{x} \sin \left (x \right )}{229205} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =2\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =5\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =6\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =3\right ) x} c_{4} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =1\right ) x} c_{5} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =4\right ) x} c_{6} -\frac {48528 \,{\mathrm e}^{x} \cos \left (x \right )}{229205}-\frac {16896 \,{\mathrm e}^{x} \sin \left (x \right )}{229205} \] 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: 6; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 6; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 5468
dsolve(diff(y(x),x$6)-12*diff(y(x),x$5)+63*diff(y(x),x$4)-18*diff(y(x),x$3)+315*diff(y(x),x$2)-300*diff(y(x),x)+125*y(x)=exp(x)*(48*cos(x)+96*sin(x)),y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 292
DSolve[y''''''[x]-12*y'''''[x]+63*y''''[x]-18*y'''[x]+315*y''[x]-300*y'[x]+125*y[x]==Exp[x]*(48*Cos[x]+96*Sin[x]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,3\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,4\right ]\right )+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,1\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,2\right ]\right )+c_5 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,5\right ]\right )+c_6 \exp \left (x \text {Root}\left [\text {$\#$1}^6-12 \text {$\#$1}^5+63 \text {$\#$1}^4-18 \text {$\#$1}^3+315 \text {$\#$1}^2-300 \text {$\#$1}+125\&,6\right ]\right )-\frac {48 e^x (352 \sin (x)+1011 \cos (x))}{229205} \]