Internal problem ID [9790]
Internal file name [OUTPUT/8733_Monday_June_06_2022_05_21_30_AM_81330792/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1464.
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]]
\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }-3 y^{\prime }+10 y=0} \] The characteristic equation is \[ \lambda ^{3}-2 \lambda ^{2}-3 \lambda +10 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -2\\ \lambda _2 &= 2-i\\ \lambda _3 &= 2+i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{\left (2-i\right ) x} c_{2} +{\mathrm e}^{\left (2+i\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-2 x}\\ y_2 &= {\mathrm e}^{\left (2-i\right ) x}\\ y_3 &= {\mathrm e}^{\left (2+i\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{\left (2-i\right ) x} c_{2} +{\mathrm e}^{\left (2+i\right ) x} c_{3} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-2 x} c_{1} +{\mathrm e}^{\left (2-i\right ) x} c_{2} +{\mathrm e}^{\left (2+i\right ) x} c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }-2 \frac {d}{d x}y^{\prime }-3 y^{\prime }+10 y=0 \\ \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 )=2 y_{3}\left (x \right )+3 y_{2}\left (x \right )-10 y_{1}\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 )=2 y_{3}\left (x \right )+3 y_{2}\left (x \right )-10 y_{1}\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 \\ -10 & 3 & 2 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -10 & 3 & 2 \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 [-2, \left [\begin {array}{c} \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]\right ], \left [2-\mathrm {I}, \left [\begin {array}{c} \frac {3}{25}+\frac {4 \,\mathrm {I}}{25} \\ \frac {2}{5}+\frac {\mathrm {I}}{5} \\ 1 \end {array}\right ]\right ], \left [2+\mathrm {I}, \left [\begin {array}{c} \frac {3}{25}-\frac {4 \,\mathrm {I}}{25} \\ \frac {2}{5}-\frac {\mathrm {I}}{5} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-2, \left [\begin {array}{c} \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{-2 x}\cdot \left [\begin {array}{c} \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [2-\mathrm {I}, \left [\begin {array}{c} \frac {3}{25}+\frac {4 \,\mathrm {I}}{25} \\ \frac {2}{5}+\frac {\mathrm {I}}{5} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (2-\mathrm {I}\right ) x}\cdot \left [\begin {array}{c} \frac {3}{25}+\frac {4 \,\mathrm {I}}{25} \\ \frac {2}{5}+\frac {\mathrm {I}}{5} \\ 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}^{2 x}\cdot \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right )\cdot \left [\begin {array}{c} \frac {3}{25}+\frac {4 \,\mathrm {I}}{25} \\ \frac {2}{5}+\frac {\mathrm {I}}{5} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{2 x}\cdot \left [\begin {array}{c} \left (\frac {3}{25}+\frac {4 \,\mathrm {I}}{25}\right ) \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ \left (\frac {2}{5}+\frac {\mathrm {I}}{5}\right ) \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ \cos \left (x \right )-\mathrm {I} \sin \left (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}^{2 x}\cdot \left [\begin {array}{c} \frac {3 \cos \left (x \right )}{25}+\frac {4 \sin \left (x \right )}{25} \\ \frac {2 \cos \left (x \right )}{5}+\frac {\sin \left (x \right )}{5} \\ \cos \left (x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{2 x}\cdot \left [\begin {array}{c} -\frac {3 \sin \left (x \right )}{25}+\frac {4 \cos \left (x \right )}{25} \\ -\frac {2 \sin \left (x \right )}{5}+\frac {\cos \left (x \right )}{5} \\ -\sin \left (x \right ) \end {array}\right ]\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}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}={\mathrm e}^{-2 x} c_{1} \cdot \left [\begin {array}{c} \frac {1}{4} \\ -\frac {1}{2} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{2 x}\cdot \left [\begin {array}{c} \frac {3 \cos \left (x \right )}{25}+\frac {4 \sin \left (x \right )}{25} \\ \frac {2 \cos \left (x \right )}{5}+\frac {\sin \left (x \right )}{5} \\ \cos \left (x \right ) \end {array}\right ]+c_{3} {\mathrm e}^{2 x}\cdot \left [\begin {array}{c} -\frac {3 \sin \left (x \right )}{25}+\frac {4 \cos \left (x \right )}{25} \\ -\frac {2 \sin \left (x \right )}{5}+\frac {\cos \left (x \right )}{5} \\ -\sin \left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (\left (3 c_{2} +4 c_{3} \right ) \cos \left (x \right )+4 \sin \left (x \right ) \left (c_{2} -\frac {3 c_{3}}{4}\right )\right ) {\mathrm e}^{2 x}}{25}+\frac {{\mathrm e}^{-2 x} c_{1}}{4} \end {array} \]
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 27
dsolve(diff(diff(diff(y(x),x),x),x)-2*diff(diff(y(x),x),x)-3*diff(y(x),x)+10*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-2 x} c_{1} +c_{2} {\mathrm e}^{2 x} \sin \left (x \right )+c_{3} {\mathrm e}^{2 x} \cos \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 34
DSolve[10*y[x] - 3*y'[x] - 2*y''[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-2 x} \left (c_2 e^{4 x} \cos (x)+c_1 e^{4 x} \sin (x)+c_3\right ) \]