23.3.23 problem 9(d)

Internal problem ID [4164]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 4. The general linear differential equation of order n. Exercises at page 63
Problem number : 9(d)
Date solved : Monday, January 27, 2025 at 08:41:09 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 9 y^{\prime \prime }-6 y^{\prime }+y&=\left (4 x^{2}+24 x +18\right ) {\mathrm e}^{x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=4 \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 21

dsolve([9*diff(y(x),x$2)-6*diff(y(x),x)+y(x)=(4*x^2+24*x+18)*exp(x),y(0) = 3, D(y)(0) = 4],y(x), singsol=all)
 

\[ y \left (x \right ) = \left (3 x +3\right ) {\mathrm e}^{\frac {x}{3}}+x^{2} {\mathrm e}^{x} \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 29

DSolve[{9*D[y[x],{x,2}]-6*D[y[x],x]+y[x]==(4*x^2+24*x+18)*Exp[x],{y[0]==3,Derivative[1][y][0] ==4}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{x/3} \left (e^{2 x/3} x^2+3 x+3\right ) \]