Internal problem ID [15275]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 15.3 Nonhomogeneous linear equations with
constant coefficients. Trial and error method. Exercises page 132
Problem number: 505.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime \prime }+2 n^{2} y^{\prime \prime }+n^{4} y=a \sin \left (n x +\alpha \right )} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 67
dsolve(diff(y(x),x$4)+2*n^2*diff(y(x),x$2)+n^4*y(x)=a*sin(n*x+alpha),y(x), singsol=all)
\[ y \left (x \right ) = \frac {a \left (-n^{2} x^{2}+2\right ) \sin \left (n x +\alpha \right )-2 \left (a x \cos \left (n x +\alpha \right )-4 \left (\left (c_{3} x +c_{1} \right ) \cos \left (n x \right )+\sin \left (n x \right ) \left (c_{4} x +c_{2} \right )\right ) n^{3}\right ) n}{8 n^{4}} \]
✓ Solution by Mathematica
Time used: 0.193 (sec). Leaf size: 79
DSolve[y''''[x]+2*n^2*y''[x]+n^4*y[x]==a*Sin[n*x+\[Alpha]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {3 a \sin (\alpha +n x)}{16 n^4}-\frac {a x \cos (\alpha +n x)}{4 n^3}-\frac {a x^2 \sin (\alpha +n x)}{8 n^2}+(c_2 x+c_1) \cos (n x)+c_4 x \sin (n x)+c_3 \sin (n x) \]