12.18.15 problem section 9.2, problem 15

Internal problem ID [2129]
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 15
Date solved : Monday, January 27, 2025 at 05:42:41 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-2 y^{\prime \prime }+4 y^{\prime }-8 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=-2\\ y^{\prime \prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 23

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

\[ y = \frac {5 \,{\mathrm e}^{2 x}}{4}-\frac {9 \sin \left (2 x \right )}{4}+\frac {3 \cos \left (2 x \right )}{4} \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 21

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

\[ y(x)\to e^{2 x}-2 \sin (2 x)+\cos (2 x) \]