Internal problem ID [11826]
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: 52.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime \prime }+16 y=x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 108
dsolve(diff(y(x),x$4)+16*y(x)=x*exp(sqrt(2)*x)*sin(sqrt(2)*x)+exp(-sqrt(2)*x)*cos(sqrt(2)*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\left (2 x \sqrt {2}+128 c_{3} +3\right ) \cos \left (x \sqrt {2}\right )+2 \sin \left (x \sqrt {2}\right ) \left (x \sqrt {2}+64 c_{4} \right )\right ) {\mathrm e}^{-x \sqrt {2}}}{128}-\frac {\left (\left (x^{2} \sqrt {2}-128 c_{1} -\frac {5 \sqrt {2}}{8}\right ) \cos \left (x \sqrt {2}\right )+\sin \left (x \sqrt {2}\right ) \left (x^{2} \sqrt {2}-3 x -128 c_{2} +\frac {5 \sqrt {2}}{8}\right )\right ) {\mathrm e}^{x \sqrt {2}}}{128} \]
✓ Solution by Mathematica
Time used: 2.857 (sec). Leaf size: 140
DSolve[y''''[x]+16*y[x]==x*Exp[Sqrt[2]*x]*Sin[Sqrt[2]*x]+Exp[-Sqrt[2]*x]*Cos[Sqrt[2]*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\sqrt {2} x} \left (\left (e^{2 \sqrt {2} x} \left (-8 \sqrt {2} x^2+5 \sqrt {2}+1024 c_1\right )+8 \left (2 \sqrt {2} x+3+128 c_2\right )\right ) \cos \left (\sqrt {2} x\right )-\left (e^{2 \sqrt {2} x} \left (8 \sqrt {2} x^2-24 x+5 \sqrt {2}-1024 c_4\right )-16 \left (\sqrt {2} x+64 c_3\right )\right ) \sin \left (\sqrt {2} x\right )\right )}{1024} \]