71.13.13 problem 13

Internal problem ID [14533]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.2, page 248
Problem number : 13
Date solved : Tuesday, January 28, 2025 at 06:43:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+2 y&=-x^{2}+1 \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: 8.183 (sec). Leaf size: 18

dsolve([diff(y(x),x$2)-2*diff(y(x),x)+2*y(x)=1-x^2,y(0) = 1, D(y)(0) = 0],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 20

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

\[ y(x)\to e^x \cos (x)-\frac {1}{2} x (x+2) \]