75.16.34 problem 507

Internal problem ID [17007]
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 : 507
Date solved : Tuesday, January 28, 2025 at 09:45:26 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+6 y^{\prime \prime }+4 y^{\prime }+y&=\sin \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 29

dsolve(diff(y(x),x$4)+4*diff(y(x),x$3)+6*diff(y(x),x$2)+4*diff(y(x),x)+y(x)=sin(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.104 (sec). Leaf size: 122

DSolve[D[y[x],{x,4}]+4*D[y[x],{x,3}]+6*D[y[x],{x,2}]+4*D[y[x],x]+y[x]==Sin[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-x} \left (x^3 \int _1^x\frac {1}{6} e^{K[4]} \sin (K[4])dK[4]+x^2 \int _1^x-\frac {1}{2} e^{K[3]} K[3] \sin (K[3])dK[3]+x \int _1^x\frac {1}{2} e^{K[2]} K[2]^2 \sin (K[2])dK[2]+\int _1^x-\frac {1}{6} e^{K[1]} K[1]^3 \sin (K[1])dK[1]+c_4 x^3+c_3 x^2+c_2 x+c_1\right ) \]