12.18.27 problem section 9.2, problem 27

Internal problem ID [2141]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.2. constant coefficient. Page 483
Problem number : section 9.2, problem 27
Date solved : Monday, January 27, 2025 at 05:42:47 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} 4 y^{\prime \prime \prime \prime }+8 y^{\prime \prime \prime }+19 y^{\prime \prime }+32 y^{\prime }+12 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=-3\\ y^{\prime \prime }\left (0\right )&=-{\frac {7}{2}}\\ y^{\prime \prime \prime }\left (0\right )&={\frac {31}{4}} \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 21

dsolve([4*diff(y(x),x$4)+8*diff(y(x),x$3)+19*diff(y(x),x$2)+32*diff(y(x),x)+12*y(x)=0,y(0) = 3, D(y)(0) = -3, (D@@2)(y)(0) = -7/2, (D@@3)(y)(0) = 31/4],y(x), singsol=all)
 

\[ y = 2 \,{\mathrm e}^{-\frac {x}{2}}-\sin \left (2 x \right )+\cos \left (2 x \right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 31

DSolve[{D[y[x],{x,4}]+2*D[y[x],{x,3}]-2*D[y[x],{x,2}]-8*D[y[x],x]-8*y[x]==0,{y[0]==5,Derivative[1][y][0] ==-2,Derivative[2][y][0] ==6,Derivative[3][y][0]==8}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-2 x} \left (e^{4 x}+e^x \sin (x)+3 e^x \cos (x)+1\right ) \]