Internal problem ID [1515]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined
Coefficients for Higher Order Equations. Page 495
Problem number: section 9.3, problem 18.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+2 y-{\mathrm e}^{2 x} \left (x^{4}+x +24\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 126
dsolve(diff(y(x),x$4)-4*diff(y(x),x$3)+6*diff(y(x),x$2)-4*diff(y(x),x)+2*y(x)=exp(2*x)*(24+x+x^4),y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (x^{4}-8 x^{3}+12 x^{2}+49 x -62\right ) \left ({\mathrm e}^{2 x} x^{4}+{\mathrm e}^{2 x} x +24 \,{\mathrm e}^{2 x}\right )}{2 x^{4}+2 x +48}+c_{1} {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}} \cos \left (\frac {\sqrt {2}\, x}{2}\right )+c_{2} {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}} \sin \left (\frac {\sqrt {2}\, x}{2}\right )+c_{3} {\mathrm e}^{-\frac {\left (\sqrt {2}-2\right ) x}{2}} \cos \left (\frac {\sqrt {2}\, x}{2}\right )+c_{4} {\mathrm e}^{-\frac {\left (\sqrt {2}-2\right ) x}{2}} \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 98
DSolve[y''''[x]-4*y'''[x]+6*y''[x]-4*y'[x]+2*y[x]==Exp[2*x]*(24+x+x^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{x-\frac {x}{\sqrt {2}}} \left (e^{\frac {x}{\sqrt {2}}+x} (x ((x-6) (x-2) x+49)-62)+2 \left (c_4 e^{\sqrt {2} x}+c_2\right ) \cos \left (\frac {x}{\sqrt {2}}\right )+2 \left (c_1 e^{\sqrt {2} x}+c_3\right ) \sin \left (\frac {x}{\sqrt {2}}\right )\right ) \\ \end{align*}