7.11.40 problem 41 and 42

Internal problem ID [361]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.5 (Nonhomogeneous equations and undetermined coefficients). Problems at page 161
Problem number : 41 and 42
Date solved : Wednesday, February 05, 2025 at 03:24:38 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }-2 y&=8 x^{5} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 47

dsolve([diff(y(x),x$4)-diff(y(x),x$3)-diff(y(x),x$2)-diff(y(x),x)-2*y(x)=8*x^5,y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(x), singsol=all)
 

\[ y = -4 x^{5}+10 x^{4}+20 x^{3}+30 x^{2}-450 x +255-96 \cos \left (x \right )+288 \sin \left (x \right )-160 \,{\mathrm e}^{-x}+{\mathrm e}^{2 x} \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 50

DSolve[{D[y[x],{x,4}]-D[y[x],{x,3}]-D[y[x],{x,2}]-D[y[x],{x,1}]-2*y[x]==8*x^5,{y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0] ==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -4 x^5+10 x^4+20 x^3+30 x^2-450 x-160 e^{-x}+e^{2 x}+288 \sin (x)-96 \cos (x)+255 \]