Internal problem ID [5361]
Internal file name [OUTPUT/4852_Sunday_February_04_2024_12_46_30_AM_84496447/index.tex]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 13. Homogeneous Linear equations with constant coefficients. Supplemetary
problems. Page 86
Problem number: 19.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+12 y^{\prime \prime }-8 y^{\prime }=0} \] The characteristic equation is \[ \lambda ^{4}-6 \lambda ^{3}+12 \lambda ^{2}-8 \lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 2\\ \lambda _3 &= 2\\ \lambda _4 &= 2 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} +c_{2} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{3} +x^{2} {\mathrm e}^{2 x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= 1\\ y_2 &= {\mathrm e}^{2 x}\\ y_3 &= {\mathrm e}^{2 x} x\\ y_4 &= x^{2} {\mathrm e}^{2 x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} +c_{2} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{3} +x^{2} {\mathrm e}^{2 x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = c_{1} +c_{2} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{3} +x^{2} {\mathrm e}^{2 x} c_{4} \] 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: 21
dsolve(diff(y(x),x$4)-6*diff(y(x),x$3)+12*diff(y(x),x$2)-8*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{4} x^{2}+c_{3} x +c_{2} \right ) {\mathrm e}^{2 x}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.03 (sec). Leaf size: 43
DSolve[y''''[x]-6*y'''[x]+12*y''[x]-8*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{4} e^{2 x} \left (c_3 \left (2 x^2-2 x+1\right )+c_2 (2 x-1)+2 c_1\right )+c_4 \]