Internal problem ID [2137]
Internal file name [OUTPUT/2137_Monday_February_26_2024_09_17_48_AM_17396529/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 18, page 82
Problem number: 21.
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+4 y^{\prime \prime }-12 y^{\prime }+16 y=0} \] The characteristic equation is \[ \lambda ^{4}-3 \lambda ^{3}+4 \lambda ^{2}-12 \lambda +16 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -\frac {1}{2}-\frac {i \sqrt {15}}{2}\\ \lambda _2 &= -\frac {1}{2}+\frac {i \sqrt {15}}{2}\\ \lambda _3 &= 2\\ \lambda _4 &= 2 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{2 x} c_{1} +x \,{\mathrm e}^{2 x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {15}}{2}\right ) x} c_{3} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {15}}{2}\right ) x} c_{4} \] 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 &= {\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {15}}{2}\right ) x}\\ y_4 &= {\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {15}}{2}\right ) 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} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {15}}{2}\right ) x} c_{3} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {15}}{2}\right ) x} c_{4} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{2 x} c_{1} +x \,{\mathrm e}^{2 x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {15}}{2}\right ) x} c_{3} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {15}}{2}\right ) x} c_{4} \] 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: 0.015 (sec). Leaf size: 41
dsolve(diff(y(x),x$4)-3*diff(y(x),x$3)+4*diff(y(x),x$2)-12*diff(y(x),x)+16*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{4} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {15}\, x}{2}\right )+c_{3} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {15}\, x}{2}\right )+{\mathrm e}^{2 x} \left (c_{2} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 57
DSolve[y''''[x]-3*y'''[x]+4*y''[x]-12*y'[x]+16*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x/2} \left (e^{5 x/2} (c_4 x+c_3)+c_2 \cos \left (\frac {\sqrt {15} x}{2}\right )+c_1 \sin \left (\frac {\sqrt {15} x}{2}\right )\right ) \]