Internal problem ID [12528]
Internal file name [OUTPUT/11181_Tuesday_October_17_2023_07_20_33_AM_47053148/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: 159.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-a^{4} y=5 a^{4} {\mathrm e}^{a x} \sin \left (a x \right )} \] 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 \[ y^{\prime \prime \prime \prime }-a^{4} y = 0 \] The characteristic equation is \[ -a^{4}+\lambda ^{4} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= a\\ \lambda _2 &= -a\\ \lambda _3 &= i a\\ \lambda _4 &= -i a \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{i a x} c_{1} +{\mathrm e}^{-i a x} c_{2} +{\mathrm e}^{-a x} c_{3} +{\mathrm e}^{a x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{i a x} \\ y_2 &= {\mathrm e}^{-i a x} \\ y_3 &= {\mathrm e}^{-a x} \\ y_4 &= {\mathrm e}^{a x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }-a^{4} y = 5 a^{4} {\mathrm e}^{a x} \sin \left (a x \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 5 a^{4} {\mathrm e}^{a x} \sin \left (a x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{a x} \cos \left (a x \right ), {\mathrm e}^{a x} \sin \left (a x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{a x}, {\mathrm e}^{i a x}, {\mathrm e}^{-a x}, {\mathrm e}^{-i a x}\} \] 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. \[ y_p = A_{1} {\mathrm e}^{a x} \cos \left (a x \right )+A_{2} {\mathrm e}^{a x} \sin \left (a x \right ) \] 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 \[ -4 A_{1} a^{4} {\mathrm e}^{a x} \cos \left (a x \right )-4 A_{2} a^{4} {\mathrm e}^{a x} \sin \left (a x \right )-a^{4} \left (A_{1} {\mathrm e}^{a x} \cos \left (a x \right )+A_{2} {\mathrm e}^{a x} \sin \left (a x \right )\right ) = 5 a^{4} {\mathrm e}^{a x} \sin \left (a x \right ) \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = 0, A_{2} = -1] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -{\mathrm e}^{a x} \sin \left (a x \right ) \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{i a x} c_{1} +{\mathrm e}^{-i a x} c_{2} +{\mathrm e}^{-a x} c_{3} +{\mathrm e}^{a x} c_{4}\right ) + \left (-{\mathrm e}^{a x} \sin \left (a x \right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{i a x} c_{1} +{\mathrm e}^{-i a x} c_{2} +{\mathrm e}^{-a x} c_{3} +{\mathrm e}^{a x} c_{4} -{\mathrm e}^{a x} \sin \left (a x \right ) \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{i a x} c_{1} +{\mathrm e}^{-i a x} c_{2} +{\mathrm e}^{-a x} c_{3} +{\mathrm e}^{a x} c_{4} -{\mathrm e}^{a x} \sin \left (a x \right ) \] 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: 0.0 (sec). Leaf size: 37
dsolve(diff(y(x),x$4)-a^4*y(x)=5*a^4*exp(a*x)*sin(a*x),y(x), singsol=all)
\[ y \left (x \right ) = c_{4} {\mathrm e}^{-a x}+\left (c_{2} -\sin \left (a x \right )\right ) {\mathrm e}^{a x}+c_{1} \cos \left (a x \right )+c_{3} \sin \left (a x \right ) \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 45
DSolve[y''''[x]-a^4*y[x]==5*a^4*Exp[a*x]*Sin[a*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 e^{-a x}+c_4 e^{a x}+c_1 \cos (a x)+\left (-e^{a x}+c_3\right ) \sin (a x) \]