75.16.33 problem 506

Internal problem ID [17006]
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 ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Trial and error method. Exercises page 132
Problem number : 506
Date solved : Tuesday, January 28, 2025 at 09:45:26 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-2 n^{2} y^{\prime \prime }+n^{4} y&=\cos \left (n x +\alpha \right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 49

dsolve(diff(y(x),x$4)-2*n^2*diff(y(x),x$2)+n^4*y(x)=cos(n*x+alpha),y(x), singsol=all)
 

\[ y = \frac {\cos \left (n x +\alpha \right )+\left (4 c_4 x +4 c_{2} \right ) n^{4} {\mathrm e}^{-n x}+\left (4 c_{3} x +4 c_{1} \right ) n^{4} {\mathrm e}^{n x}}{4 n^{4}} \]

✓ Solution by Mathematica

Time used: 0.168 (sec). Leaf size: 177

DSolve[D[y[x],{x,4}]-2*n^2*D[y[x],{x,2}]+n^4*y[x]==Cos[n*x+\[Alpha]],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-n x} \left (\int _1^x-\frac {e^{n K[1]} \cos (\alpha +n K[1]) (n K[1]-1)}{4 n^3}dK[1]+e^{2 n x} \int _1^x-\frac {e^{-n K[3]} \cos (\alpha +n K[3]) (n K[3]+1)}{4 n^3}dK[3]+x \int _1^x\frac {e^{n K[2]} \cos (\alpha +n K[2])}{4 n^2}dK[2]+x e^{2 n x} \int _1^x\frac {e^{-n K[4]} \cos (\alpha +n K[4])}{4 n^2}dK[4]+c_3 e^{2 n x}+c_4 x e^{2 n x}+c_2 x+c_1\right ) \]