71.12.4 problem 4

Internal problem ID [14520]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.5, page 221
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 06:42:55 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y&=3 x +4 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=1\\ y^{\prime \prime \prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 24

dsolve([diff(y(x),x$4)+2*diff(y(x),x$2)+y(x)=3*x+4,y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 1, (D@@3)(y)(0) = 1],y(x), singsol=all)
 

\[ y = 4+\left (x -4\right ) \cos \left (x \right )+\frac {\left (-3 x -8\right ) \sin \left (x \right )}{2}+3 x \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 27

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

\[ y(x)\to 3 x-\frac {1}{2} (3 x+8) \sin (x)+(x-4) \cos (x)+4 \]