Internal problem ID [12516]
Internal file name [OUTPUT/11169_Monday_October_16_2023_09_54_23_PM_29243652/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: 147.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-a^{4} y=0} \] 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*}
Summary
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} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{i a x} c_{1} +{\mathrm e}^{-i a x} c_{2} +{\mathrm e}^{-a x} c_{3} +{\mathrm e}^{a x} c_{4} \] Verified OK.
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: 30
dsolve(diff(y(x),x$4)-a^4*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{a x} c_{1} +c_{2} {\mathrm e}^{-a x}+c_{3} \sin \left (a x \right )+c_{4} \cos \left (a x \right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 37
DSolve[y''''[x]-a^4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 e^{-a x}+c_4 e^{a x}+c_1 \cos (a x)+c_3 \sin (a x) \]