Internal problem ID [13557]
Internal file name [OUTPUT/12729_Saturday_February_17_2024_08_44_13_AM_97534733/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 14. Higher order equations and the reduction of order method. Additional
exercises page 277
Problem number: 14.5 (c).
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+24 y^{\prime \prime }-32 y^{\prime }+16 y=0} \] The characteristic equation is \[ \lambda ^{4}-8 \lambda ^{3}+24 \lambda ^{2}-32 \lambda +16 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 2\\ \lambda _2 &= 2\\ \lambda _3 &= 2\\ \lambda _4 &= 2 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{2 x} c_{1} +x \,{\mathrm e}^{2 x} c_{2} +x^{2} {\mathrm e}^{2 x} c_{3} +x^{3} {\mathrm e}^{2 x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{2 x}\\ y_2 &= x \,{\mathrm e}^{2 x}\\ y_3 &= x^{2} {\mathrm e}^{2 x}\\ y_4 &= x^{3} {\mathrm e}^{2 x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{2 x} c_{1} +x \,{\mathrm e}^{2 x} c_{2} +x^{2} {\mathrm e}^{2 x} c_{3} +x^{3} {\mathrm e}^{2 x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{2 x} c_{1} +x \,{\mathrm e}^{2 x} c_{2} +x^{2} {\mathrm e}^{2 x} c_{3} +x^{3} {\mathrm e}^{2 x} c_{4} \] 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: 24
dsolve([diff(y(x),x$4)-8*diff(y(x),x$3)+24*diff(y(x),x$2)-32*diff(y(x),x)+16*y(x)=0,exp(2*x)],singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (c_{4} x^{3}+x^{2} c_{3} +c_{2} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 28
DSolve[y''''[x]-8*y'''[x]+24*y''[x]-32*y'[x]+16*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{2 x} (x (x (c_4 x+c_3)+c_2)+c_1) \]