71.14.6 problem 12

Internal problem ID [14540]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.3, page 255
Problem number : 12
Date solved : Tuesday, January 28, 2025 at 06:43:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-y^{\prime }-2 y&=x^{2} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&={\frac {11}{4}}\\ y^{\prime }\left (0\right )&={\frac {1}{2}} \end{align*}

✓ Solution by Maple

Time used: 8.176 (sec). Leaf size: 26

dsolve([diff(y(x),x$2)-diff(y(x),x)-2*y(x)=x^2,y(0) = 11/4, D(y)(0) = 1/2],y(x), singsol=all)
 

\[ y = -\frac {x^{2}}{2}+\frac {x}{2}+\frac {7 \,{\mathrm e}^{-x}}{3}-\frac {3}{4}+\frac {7 \,{\mathrm e}^{2 x}}{6} \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 33

DSolve[{D[y[x],{x,2}]-D[y[x],x]-2*y[x]==x^2,{y[0]==11/4,Derivative[1][y][0] ==1/2}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{12} \left (-6 x^2+6 x+28 e^{-x}+14 e^{2 x}-9\right ) \]