71.15.7 problem 4 (h)

Internal problem ID [14549]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.4, page 265
Problem number : 4 (h)
Date solved : Tuesday, January 28, 2025 at 06:43:17 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+5 y&=\left \{\begin {array}{cc} x & 0\le x <1 \\ 1 & 1\le x \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 13.729 (sec). Leaf size: 85

dsolve([diff(y(x),x$2)-4*diff(y(x),x)+5*y(x)=piecewise(0<=x and x<1,x,1<=x,1),y(0) = 1, D(y)(0) = 0],y(x), singsol=all)
 

\[ y = \frac {\left (\left \{\begin {array}{cc} 4+5 x +{\mathrm e}^{2 x} \left (21 \cos \left (x \right )-47 \sin \left (x \right )\right ) & x <1 \\ 10+{\mathrm e}^{2} \left (21 \cos \left (1\right )-47 \sin \left (1\right )\right ) & x =1 \\ \left (4 \cos \left (x -1\right )-3 \sin \left (x -1\right )\right ) {\mathrm e}^{2 x -2}+5+{\mathrm e}^{2 x} \left (21 \cos \left (x \right )-47 \sin \left (x \right )\right ) & 1<x \end {array}\right .\right )}{25} \]

✓ Solution by Mathematica

Time used: 0.049 (sec). Leaf size: 119

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

\[ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{2 x} (\cos (x)-2 \sin (x)) & x\leq 0 \\ \frac {1}{25} \left (5 x+21 e^{2 x} \cos (x)-47 e^{2 x} \sin (x)+4\right ) & 0<x\leq 1 \\ \frac {4 e^{2 x} \cos (1-x)+21 e^{2 x+2} \cos (x)+3 e^{2 x} \sin (1-x)-47 e^{2 x+2} \sin (x)+5 e^2}{25 e^2} & \text {True} \\ \end {array} \\ \end {array} \]