Internal problem ID [11250]
Internal file name [OUTPUT/10236_Sunday_December_11_2022_01_27_02_AM_91030160/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter VII, Linear differential equations with constant coefficients. Article 45. Roots of
auxiliary equation complex. Page 95
Problem number: Ex 2.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y=0} \] The characteristic equation is \[ \lambda ^{4}+2 \lambda ^{2}+1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= i\\ \lambda _2 &= -i\\ \lambda _3 &= i\\ \lambda _4 &= -i \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{i x} c_{1} +x \,{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} +x \,{\mathrm e}^{-i x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{i x}\\ y_2 &= x \,{\mathrm e}^{i x}\\ y_3 &= {\mathrm e}^{-i x}\\ y_4 &= x \,{\mathrm e}^{-i x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{i x} c_{1} +x \,{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} +x \,{\mathrm e}^{-i x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{i x} c_{1} +x \,{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} +x \,{\mathrm e}^{-i x} c_{4} \] 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: 21
dsolve(diff(y(x),x$4)+2*diff(y(x),x$2)+y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{4} x +c_{2} \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_{3} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 26
DSolve[y''''[x]+2*y''[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (c_2 x+c_1) \cos (x)+(c_4 x+c_3) \sin (x) \]