Internal problem ID [13820]
Internal file name [OUTPUT/12992_Sunday_February_18_2024_08_02_34_AM_8996891/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 25. Review exercises for part III. page 447
Problem number: 23.
ODE order: 3.
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
[[_3rd_order, _missing_x]]
\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }=8} \] 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 \[ y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime } = 0 \] The characteristic equation is \[ \lambda ^{3}-6 \lambda ^{2}+12 \lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 3+i \sqrt {3}\\ \lambda _3 &= 3-i \sqrt {3} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} +{\mathrm e}^{\left (3+i \sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (3-i \sqrt {3}\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= {\mathrm e}^{\left (3+i \sqrt {3}\right ) x} \\ y_3 &= {\mathrm e}^{\left (3-i \sqrt {3}\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime } = 8 \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 1 \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \left \{1, {\mathrm e}^{\left (3-i \sqrt {3}\right ) x}, {\mathrm e}^{\left (3+i \sqrt {3}\right ) x}\right \} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x\}] \] Since there was duplication between the basis functions in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis function in the above updated UC_set. \[ y_p = A_{1} x \] The unknowns \(\{A_{1}\}\) 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 \[ 12 A_{1} = 8 \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = {\frac {2}{3}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {2 x}{3} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} +{\mathrm e}^{\left (3+i \sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (3-i \sqrt {3}\right ) x} c_{3}\right ) + \left (\frac {2 x}{3}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} +{\mathrm e}^{\left (3+i \sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (3-i \sqrt {3}\right ) x} c_{3} +\frac {2 x}{3} \\ \end{align*}
Verification of solutions
\[ y = c_{1} +{\mathrm e}^{\left (3+i \sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (3-i \sqrt {3}\right ) x} c_{3} +\frac {2 x}{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }-6 \frac {d}{d x}y^{\prime }+12 y^{\prime }=8 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \\ \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 )=\frac {d}{d x}y^{\prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{3}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{3}^{\prime }\left (x \right )=8+6 y_{3}\left (x \right )-12 y_{2}\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_{3}^{\prime }\left (x \right )=8+6 y_{3}\left (x \right )-12 y_{2}\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 ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & -12 & 6 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right )+\left [\begin {array}{c} 0 \\ 0 \\ 8 \end {array}\right ] \\ \bullet & {} & \textrm {Define the forcing function}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{f}}\left (x \right )=\left [\begin {array}{c} 0 \\ 0 \\ 8 \end {array}\right ] \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & -12 & 6 \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 )+{\moverset {\rightarrow }{f}} \\ \bullet & {} & \textrm {To solve the system, find the eigenvalues and eigenvectors of}\hspace {3pt} A \\ \bullet & {} & \textrm {Eigenpairs of}\hspace {3pt} A \\ {} & {} & \left [\left [0, \left [\begin {array}{c} 1 \\ 0 \\ 0 \end {array}\right ]\right ], \left [3-\mathrm {I} \sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (3-\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {1}{3-\mathrm {I} \sqrt {3}} \\ 1 \end {array}\right ]\right ], \left [3+\mathrm {I} \sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (3+\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {1}{3+\mathrm {I} \sqrt {3}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [0, \left [\begin {array}{c} 1 \\ 0 \\ 0 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}=\left [\begin {array}{c} 1 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [3-\mathrm {I} \sqrt {3}, \left [\begin {array}{c} \frac {1}{\left (3-\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {1}{3-\mathrm {I} \sqrt {3}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (3-\mathrm {I} \sqrt {3}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (3-\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {1}{3-\mathrm {I} \sqrt {3}} \\ 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 \\ {} & {} & {\mathrm e}^{3 x}\cdot \left (\cos \left (\sqrt {3}\, x \right )-\mathrm {I} \sin \left (\sqrt {3}\, x \right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (3-\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {1}{3-\mathrm {I} \sqrt {3}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {3}\, x \right )-\mathrm {I} \sin \left (\sqrt {3}\, x \right )}{\left (3-\mathrm {I} \sqrt {3}\right )^{2}} \\ \frac {\cos \left (\sqrt {3}\, x \right )-\mathrm {I} \sin \left (\sqrt {3}\, x \right )}{3-\mathrm {I} \sqrt {3}} \\ \cos \left (\sqrt {3}\, x \right )-\mathrm {I} \sin \left (\sqrt {3}\, x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{2}\left (x \right )={\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {3}\, x \right )}{24}+\frac {\sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{24} \\ \frac {\cos \left (\sqrt {3}\, x \right )}{4}+\frac {\sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{12} \\ \cos \left (\sqrt {3}\, x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {3}}{24}-\frac {\sin \left (\sqrt {3}\, x \right )}{24} \\ \frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {3}}{12}-\frac {\sin \left (\sqrt {3}\, x \right )}{4} \\ -\sin \left (\sqrt {3}\, x \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {General solution of the system of ODEs can be written in terms of the particular solution}\hspace {3pt} {\moverset {\rightarrow }{y}}_{p}\left (x \right ) \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=c_{1} {\moverset {\rightarrow }{y}}_{1}+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right )+{\moverset {\rightarrow }{y}}_{p}\left (x \right ) \\ \square & {} & \textrm {Fundamental matrix}\hspace {3pt} \\ {} & \circ & \textrm {Let}\hspace {3pt} \phi \left (x \right )\hspace {3pt}\textrm {be the matrix whose columns are the independent solutions of the homogeneous system.}\hspace {3pt} \\ {} & {} & \phi \left (x \right )=\left [\begin {array}{ccc} 1 & {\mathrm e}^{3 x} \left (\frac {\cos \left (\sqrt {3}\, x \right )}{24}+\frac {\sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{24}\right ) & {\mathrm e}^{3 x} \left (\frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {3}}{24}-\frac {\sin \left (\sqrt {3}\, x \right )}{24}\right ) \\ 0 & {\mathrm e}^{3 x} \left (\frac {\cos \left (\sqrt {3}\, x \right )}{4}+\frac {\sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{12}\right ) & {\mathrm e}^{3 x} \left (\frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {3}}{12}-\frac {\sin \left (\sqrt {3}\, x \right )}{4}\right ) \\ 0 & {\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right ) & -{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \end {array}\right ] \\ {} & \circ & \textrm {The fundamental matrix,}\hspace {3pt} \Phi \left (x \right )\hspace {3pt}\textrm {is a normalized version of}\hspace {3pt} \phi \left (x \right )\hspace {3pt}\textrm {satisfying}\hspace {3pt} \Phi \left (0\right )=I \hspace {3pt}\textrm {where}\hspace {3pt} I \hspace {3pt}\textrm {is the identity matrix}\hspace {3pt} \\ {} & {} & \Phi \left (x \right )=\phi \left (x \right )\cdot \frac {1}{\phi \left (0\right )} \\ {} & \circ & \textrm {Substitute the value of}\hspace {3pt} \phi \left (x \right )\hspace {3pt}\textrm {and}\hspace {3pt} \phi \left (0\right ) \\ {} & {} & \Phi \left (x \right )=\left [\begin {array}{ccc} 1 & {\mathrm e}^{3 x} \left (\frac {\cos \left (\sqrt {3}\, x \right )}{24}+\frac {\sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{24}\right ) & {\mathrm e}^{3 x} \left (\frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {3}}{24}-\frac {\sin \left (\sqrt {3}\, x \right )}{24}\right ) \\ 0 & {\mathrm e}^{3 x} \left (\frac {\cos \left (\sqrt {3}\, x \right )}{4}+\frac {\sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{12}\right ) & {\mathrm e}^{3 x} \left (\frac {\cos \left (\sqrt {3}\, x \right ) \sqrt {3}}{12}-\frac {\sin \left (\sqrt {3}\, x \right )}{4}\right ) \\ 0 & {\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right ) & -{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{ccc} 1 & \frac {1}{24} & \frac {\sqrt {3}}{24} \\ 0 & \frac {1}{4} & \frac {\sqrt {3}}{12} \\ 0 & 1 & 0 \end {array}\right ]} \\ {} & \circ & \textrm {Evaluate and simplify to get the fundamental matrix}\hspace {3pt} \\ {} & {} & \Phi \left (x \right )=\left [\begin {array}{ccc} 1 & \frac {{\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right )}{2}-\frac {{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{6}-\frac {1}{2} & -\frac {{\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right )}{12}+\frac {{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{12}+\frac {1}{12} \\ 0 & \frac {\left (\cos \left (\sqrt {3}\, x \right ) \sqrt {3}-3 \sin \left (\sqrt {3}\, x \right )\right ) \sqrt {3}\, {\mathrm e}^{3 x}}{3} & \frac {{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{3} \\ 0 & -4 \,{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \sqrt {3} & {\mathrm e}^{3 x} \left (\cos \left (\sqrt {3}\, x \right )+\sin \left (\sqrt {3}\, x \right ) \sqrt {3}\right ) \end {array}\right ] \\ \square & {} & \textrm {Find a particular solution of the system of ODEs using variation of parameters}\hspace {3pt} \\ {} & \circ & \textrm {Let the particular solution be the fundamental matrix multiplied by}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right )\hspace {3pt}\textrm {and solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right ) \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (x \right )=\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right ) \\ {} & \circ & \textrm {Take the derivative of the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}^{\prime }\left (x \right )=\Phi ^{\prime }\left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right ) \\ {} & \circ & \textrm {Substitute particular solution and its derivative into the system of ODEs}\hspace {3pt} \\ {} & {} & \Phi ^{\prime }\left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )=A \cdot \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+{\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {The fundamental matrix has columns that are solutions to the homogeneous system so its derivative follows that of the homogeneous system}\hspace {3pt} \\ {} & {} & A \cdot \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )=A \cdot \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+{\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {Cancel like terms}\hspace {3pt} \\ {} & {} & \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )={\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {Multiply by the inverse of the fundamental matrix}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )=\frac {1}{\Phi \left (x \right )}\cdot {\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {Integrate to solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right ) \\ {} & {} & {\moverset {\rightarrow }{v}}\left (x \right )=\int _{0}^{x}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \\ {} & \circ & \textrm {Plug}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right )\hspace {3pt}\textrm {into the equation for the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (x \right )=\Phi \left (x \right )\cdot \left (\int _{0}^{x}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \right ) \\ {} & \circ & \textrm {Plug in the fundamental matrix and the forcing function and compute}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (x \right )=\left [\begin {array}{c} \frac {{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{9}-\frac {{\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right )}{3}+\frac {1}{3}+\frac {2 x}{3} \\ \frac {2 \,{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{3}-\frac {2 \,{\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right )}{3}+\frac {2}{3} \\ \frac {8 \,{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{3} \end {array}\right ] \\ \bullet & {} & \textrm {Plug particular solution back into general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=c_{1} {\moverset {\rightarrow }{y}}_{1}+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right )+\left [\begin {array}{c} \frac {{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{9}-\frac {{\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right )}{3}+\frac {1}{3}+\frac {2 x}{3} \\ \frac {2 \,{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{3}-\frac {2 \,{\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right )}{3}+\frac {2}{3} \\ \frac {8 \,{\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right ) \sqrt {3}}{3} \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{3 x} \left (c_{3} \sqrt {3}+c_{2} -8\right ) \cos \left (\sqrt {3}\, x \right )}{24}+\frac {\left (\left (c_{2} +\frac {8}{3}\right ) \sqrt {3}-c_{3} \right ) {\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right )}{24}+\frac {2 x}{3}+c_{1} +\frac {1}{3} \end {array} \]
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 3; linear nonhomogeneous with symmetry [0,1] -> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = 6*(diff(_b(_a), _a))-12*_b(_a)+8, _b(_a)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients <- constant coefficients successful <- solving first the homogeneous part of the ODE successful <- differential order: 3; linear nonhomogeneous with symmetry [0,1] successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 50
dsolve(diff(y(x),x$3)-6*diff(y(x),x$2)+12*diff(y(x),x)=8,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-\frac {\sqrt {3}\, c_{2}}{3}+c_{1} \right ) {\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right )}{4}+\frac {{\mathrm e}^{3 x} \left (\sqrt {3}\, c_{1} +3 c_{2} \right ) \sin \left (\sqrt {3}\, x \right )}{12}+\frac {2 x}{3}+c_{3} \]
✓ Solution by Mathematica
Time used: 0.312 (sec). Leaf size: 71
DSolve[y'''[x]-6*y''[x]+12*y'[x]==8,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{12} \left (8 x-\left (\sqrt {3} c_1-3 c_2\right ) e^{3 x} \cos \left (\sqrt {3} x\right )+\left (3 c_1+\sqrt {3} c_2\right ) e^{3 x} \sin \left (\sqrt {3} x\right )\right )+c_3 \]