64.11.47 problem 47

Internal problem ID [13497]
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 : 47
Date solved : Tuesday, January 28, 2025 at 05:48:48 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y&=x \,{\mathrm e}^{2 x}+x^{2} {\mathrm e}^{3 x} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 41

dsolve(diff(y(x),x$3)-6*diff(y(x),x$2)+12*diff(y(x),x)-8*y(x)=x*exp(2*x)+x^2*exp(3*x),y(x), singsol=all)
 

\[ y = \frac {\left (x^{4}+24 c_{3} x^{2}+24 c_{2} x +24 c_{1} \right ) {\mathrm e}^{2 x}}{24}+{\mathrm e}^{3 x} \left (x^{2}-6 x +12\right ) \]

✓ Solution by Mathematica

Time used: 0.149 (sec). Leaf size: 98

DSolve[D[y[x],{x,3}]-6*D[y[x],{x,2}]+12*D[y[x],x]-8*y[x]==x*Exp[2*x]+x^2*Exp[3*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{2 x} \left (x^2 \int _1^x\frac {1}{2} K[3] \left (e^{K[3]} K[3]+1\right )dK[3]+x \int _1^x-K[2]^2 \left (e^{K[2]} K[2]+1\right )dK[2]+\int _1^x\frac {1}{2} K[1]^3 \left (e^{K[1]} K[1]+1\right )dK[1]+c_3 x^2+c_2 x+c_1\right ) \]