75.16.68 problem 541

Internal problem ID [17041]
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 : 541
Date solved : Tuesday, January 28, 2025 at 09:47:23 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }+2 y^{\prime \prime }-2 y^{\prime }+y&={\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 89

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

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