Internal problem ID [304]
Internal file name [OUTPUT/304_Sunday_June_05_2022_01_38_26_AM_40263199/index.tex]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 5.3, Higher-Order Linear Differential Equations. Homogeneous Equations with
Constant Coefficients. Page 300
Problem number: problem 32.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }-3 y^{\prime \prime }-5 y^{\prime }-2 y=0} \] The characteristic equation is \[ \lambda ^{4}+\lambda ^{3}-3 \lambda ^{2}-5 \lambda -2 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 2\\ \lambda _2 &= -1\\ \lambda _3 &= -1\\ \lambda _4 &= -1 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-x}+x \,{\mathrm e}^{-x} c_{2} +x^{2} {\mathrm e}^{-x} c_{3} +{\mathrm e}^{2 x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-x}\\ y_2 &= x \,{\mathrm e}^{-x}\\ y_3 &= x^{2} {\mathrm e}^{-x}\\ y_4 &= {\mathrm e}^{2 x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-x}+x \,{\mathrm e}^{-x} c_{2} +x^{2} {\mathrm e}^{-x} c_{3} +{\mathrm e}^{2 x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{-x}+x \,{\mathrm e}^{-x} c_{2} +x^{2} {\mathrm e}^{-x} c_{3} +{\mathrm e}^{2 x} c_{4} \] 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: 26
dsolve(diff(y(x),x$4)+diff(y(x),x$3)-3*diff(y(x),x$2)-5*diff(y(x),x)-2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{4} x^{2}+c_{3} x +c_{2} \right ) {\mathrm e}^{-x}+{\mathrm e}^{2 x} c_{1} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 32
DSolve[y''''[x]+y'''[x]-3*y''[x]-5*y'[x]-2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x} \left (c_3 x^2+c_2 x+c_4 e^{3 x}+c_1\right ) \]