Internal problem ID [12513]
Internal file name [OUTPUT/11166_Monday_October_16_2023_09_54_23_PM_42698315/index.tex]
Book: DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR
PUBLISHERS, Moscow 1969.
Section: Chapter 8. Differential equations. Exercises page 595
Problem number: 144.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+9 y=0} \] The characteristic equation is \[ \lambda ^{4}+2 \lambda ^{2}+9 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -1-i \sqrt {2}\\ \lambda _2 &= -1+i \sqrt {2}\\ \lambda _3 &= 1-i \sqrt {2}\\ \lambda _4 &= 1+i \sqrt {2} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (1-i \sqrt {2}\right ) x} c_{1} +{\mathrm e}^{\left (-1-i \sqrt {2}\right ) x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {2}\right ) x} c_{3} +{\mathrm e}^{\left (-1+i \sqrt {2}\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (1-i \sqrt {2}\right ) x}\\ y_2 &= {\mathrm e}^{\left (-1-i \sqrt {2}\right ) x}\\ y_3 &= {\mathrm e}^{\left (1+i \sqrt {2}\right ) x}\\ y_4 &= {\mathrm e}^{\left (-1+i \sqrt {2}\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (1-i \sqrt {2}\right ) x} c_{1} +{\mathrm e}^{\left (-1-i \sqrt {2}\right ) x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {2}\right ) x} c_{3} +{\mathrm e}^{\left (-1+i \sqrt {2}\right ) x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\left (1-i \sqrt {2}\right ) x} c_{1} +{\mathrm e}^{\left (-1-i \sqrt {2}\right ) x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {2}\right ) x} c_{3} +{\mathrm e}^{\left (-1+i \sqrt {2}\right ) x} c_{4} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+9 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & y^{\prime \prime \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 )=y^{\prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{4}\left (x \right ) \\ {} & {} & y_{4}\left (x \right )=y^{\prime \prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{4}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{4}^{\prime }\left (x \right )=-2 y_{3}\left (x \right )-9 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_{4}\left (x \right )=y_{3}^{\prime }\left (x \right ), y_{4}^{\prime }\left (x \right )=-2 y_{3}\left (x \right )-9 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 ) \\ y_{4}\left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -9 & 0 & -2 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -9 & 0 & -2 & 0 \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 [-1-\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (-1-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (-1-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{-1-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ], \left [-1+\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (-1+\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (-1+\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{-1+\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ], \left [1-\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (1-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (1-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{1-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ], \left [1+\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (1+\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (1+\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{1+\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-1-\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (-1-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (-1-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{-1-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-1-\mathrm {I} \sqrt {2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-1-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (-1-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{-1-\mathrm {I} \sqrt {2}} \\ 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}^{-x}\cdot \left (\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-1-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (-1-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{-1-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{\left (-1-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{\left (-1-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{-1-\mathrm {I} \sqrt {2}} \\ \cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{1}\left (x \right )={\mathrm e}^{-x}\cdot \left [\begin {array}{c} \frac {5 \cos \left (\sqrt {2}\, x \right )}{27}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{27} \\ -\frac {\cos \left (\sqrt {2}\, x \right )}{9}-\frac {2 \sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{9} \\ -\frac {\cos \left (\sqrt {2}\, x \right )}{3}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{3} \\ \cos \left (\sqrt {2}\, x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{2}\left (x \right )={\mathrm e}^{-x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{27}-\frac {5 \sin \left (\sqrt {2}\, x \right )}{27} \\ -\frac {2 \cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{9}+\frac {\sin \left (\sqrt {2}\, x \right )}{9} \\ \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{3}+\frac {\sin \left (\sqrt {2}\, x \right )}{3} \\ -\sin \left (\sqrt {2}\, x \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [1-\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (1-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (1-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{1-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (1-\mathrm {I} \sqrt {2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (1-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (1-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{1-\mathrm {I} \sqrt {2}} \\ 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}^{x}\cdot \left (\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (1-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (1-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{1-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{\left (1-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{\left (1-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{1-\mathrm {I} \sqrt {2}} \\ \cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{x}\cdot \left [\begin {array}{c} -\frac {5 \cos \left (\sqrt {2}\, x \right )}{27}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{27} \\ -\frac {\cos \left (\sqrt {2}\, x \right )}{9}+\frac {2 \sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{9} \\ \frac {\cos \left (\sqrt {2}\, x \right )}{3}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{3} \\ \cos \left (\sqrt {2}\, x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{27}+\frac {5 \sin \left (\sqrt {2}\, x \right )}{27} \\ \frac {2 \cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{9}+\frac {\sin \left (\sqrt {2}\, x \right )}{9} \\ \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{3}-\frac {\sin \left (\sqrt {2}\, x \right )}{3} \\ -\sin \left (\sqrt {2}\, 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}\left (x \right )+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (x \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}={\mathrm e}^{-x} c_{1} \cdot \left [\begin {array}{c} \frac {5 \cos \left (\sqrt {2}\, x \right )}{27}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{27} \\ -\frac {\cos \left (\sqrt {2}\, x \right )}{9}-\frac {2 \sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{9} \\ -\frac {\cos \left (\sqrt {2}\, x \right )}{3}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{3} \\ \cos \left (\sqrt {2}\, x \right ) \end {array}\right ]+c_{2} {\mathrm e}^{-x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{27}-\frac {5 \sin \left (\sqrt {2}\, x \right )}{27} \\ -\frac {2 \cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{9}+\frac {\sin \left (\sqrt {2}\, x \right )}{9} \\ \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{3}+\frac {\sin \left (\sqrt {2}\, x \right )}{3} \\ -\sin \left (\sqrt {2}\, x \right ) \end {array}\right ]+c_{3} {\mathrm e}^{x}\cdot \left [\begin {array}{c} -\frac {5 \cos \left (\sqrt {2}\, x \right )}{27}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{27} \\ -\frac {\cos \left (\sqrt {2}\, x \right )}{9}+\frac {2 \sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{9} \\ \frac {\cos \left (\sqrt {2}\, x \right )}{3}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{3} \\ \cos \left (\sqrt {2}\, x \right ) \end {array}\right ]+c_{4} {\mathrm e}^{x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{27}+\frac {5 \sin \left (\sqrt {2}\, x \right )}{27} \\ \frac {2 \cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{9}+\frac {\sin \left (\sqrt {2}\, x \right )}{9} \\ \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{3}-\frac {\sin \left (\sqrt {2}\, x \right )}{3} \\ -\sin \left (\sqrt {2}\, x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (\left (c_{2} \sqrt {2}+5 c_{1} \right ) {\mathrm e}^{-x}-5 \left (-\frac {c_{4} \sqrt {2}}{5}+c_{3} \right ) {\mathrm e}^{x}\right ) \cos \left (\sqrt {2}\, x \right )}{27}+\frac {\left (\left (c_{1} \sqrt {2}-5 c_{2} \right ) {\mathrm e}^{-x}+{\mathrm e}^{x} \left (\sqrt {2}\, c_{3} +5 c_{4} \right )\right ) \sin \left (\sqrt {2}\, x \right )}{27} \end {array} \]
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: 0.0 (sec). Leaf size: 41
dsolve(diff(y(x),x$4)+2*diff(y(x),x$2)+9*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{x}+c_{4} {\mathrm e}^{-x}\right ) \cos \left (x \sqrt {2}\right )+\sin \left (x \sqrt {2}\right ) \left (c_{1} {\mathrm e}^{x}+c_{3} {\mathrm e}^{-x}\right ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 52
DSolve[y''''[x]+2*y''[x]+9*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x} \left (\left (c_4 e^{2 x}+c_2\right ) \cos \left (\sqrt {2} x\right )+\left (c_3 e^{2 x}+c_1\right ) \sin \left (\sqrt {2} x\right )\right ) \]