Internal problem ID [11822]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page
151
Problem number: 48.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }+4 y^{\prime \prime }+3 y^{\prime }+y=x^{2} {\mathrm e}^{-x}+3 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 73
dsolve(diff(y(x),x$4)+3*diff(y(x),x$3)+4*diff(y(x),x$2)+3*diff(y(x),x)+y(x)=x^2*exp(-x)+3*exp(-x/2)*cos(sqrt(3)/2*x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {3 \,{\mathrm e}^{-\frac {x}{2}} \left (x -\frac {2 c_{3}}{3}+\frac {1}{3}\right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {{\mathrm e}^{-\frac {x}{2}} \left (\left (x -5\right ) \sqrt {3}-2 c_{4} \right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {\left (-24+x^{4}+4 x^{3}+12 \left (-2+c_{2} \right ) x +12 c_{1} \right ) {\mathrm e}^{-x}}{12} \]
✓ Solution by Mathematica
Time used: 2.054 (sec). Leaf size: 104
DSolve[y''''[x]+3*y'''[x]+4*y''[x]+3*y'[x]+y[x]==x^2*Exp[-x]+3*Exp[-x/2]*Cos[Sqrt[3]/2*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{12} e^{-x} \left (x^4+4 x^3-24 x+12 c_4 x-6 e^{x/2} (3 x+1-2 c_2) \cos \left (\frac {\sqrt {3} x}{2}\right )-6 e^{x/2} \left (\sqrt {3} x-5 \sqrt {3}-2 c_1\right ) \sin \left (\frac {\sqrt {3} x}{2}\right )-24+12 c_3\right ) \]