75.16.32 problem 505

Internal problem ID [17005]
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 : 505
Date solved : Tuesday, January 28, 2025 at 09:45:25 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.010 (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)
 

\[ y = \frac {a \left (-x^{2} n^{2}+2\right ) \sin \left (n x +\alpha \right )-2 \left (a x \cos \left (n x +\alpha \right )-4 \left (\left (c_{3} x +c_{1} \right ) \cos \left (n x \right )+\sin \left (n x \right ) \left (c_4 x +c_{2} \right )\right ) n^{3}\right ) n}{8 n^{4}} \]

✓ Solution by Mathematica

Time used: 0.427 (sec). Leaf size: 188

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

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