64.11.41 problem 41

Internal problem ID [13491]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 41
Date solved : Tuesday, January 28, 2025 at 05:45:47 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+8 y&=x^{3}+x +{\mathrm e}^{-2 x} \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 38

dsolve(diff(y(x),x$2)-6*diff(y(x),x)+8*y(x)=x^3+x+exp(-2*x),y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{4 x} c_{1}}{2}+\frac {{\mathrm e}^{-2 x}}{24}+\frac {69}{256}+\frac {29 x}{64}+\frac {9 x^{2}}{32}+\frac {x^{3}}{8}+{\mathrm e}^{2 x} c_{2} \]

✓ Solution by Mathematica

Time used: 0.738 (sec). Leaf size: 96

DSolve[D[y[x],{x,2}]-6*D[y[x],x]+8*y[x]==x^3+x+Exp[-2*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{2 x} \left (\int _1^x\frac {1}{2} e^{-4 K[1]} \left (-e^{2 K[1]} K[1] \left (K[1]^2+1\right )-1\right )dK[1]+e^{2 x} \int _1^x\frac {1}{2} e^{-6 K[2]} \left (e^{2 K[2]} \left (K[2]^3+K[2]\right )+1\right )dK[2]+c_2 e^{2 x}+c_1\right ) \]