71.13.6 problem 6

Internal problem ID [14526]
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 : 6
Date solved : Tuesday, January 28, 2025 at 06:42:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=x \,{\mathrm e}^{x}-3 x^{2} \end{align*}

Using Laplace method

✓ Solution by Maple

Time used: 10.411 (sec). Leaf size: 52

dsolve(diff(y(x),x$2)+diff(y(x),x)-2*y(x)=x*exp(x)-3*x^2,y(x), singsol=all)
 

\[ y = \frac {9}{4}+\frac {3 x^{2}}{2}+\frac {3 x}{2}+\frac {{\mathrm e}^{x} \left (9 x^{2}+18 y^{\prime }\left (0\right )+36 y \left (0\right )-6 x -106\right )}{54}+\frac {\left (36 y \left (0\right )-36 y^{\prime }\left (0\right )-31\right ) {\mathrm e}^{-2 x}}{108} \]

✓ Solution by Mathematica

Time used: 0.381 (sec). Leaf size: 81

DSolve[D[y[x],{x,2}]+D[y[x],x]-2*y[x]==x*Exp[x]-3*x^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-2 x} \left (\int _1^x-\frac {1}{3} e^{2 K[1]} \left (e^{K[1]}-3 K[1]\right ) K[1]dK[1]+e^{3 x} \int _1^x\frac {1}{3} K[2] \left (1-3 e^{-K[2]} K[2]\right )dK[2]+c_2 e^{3 x}+c_1\right ) \]