23.3.18 problem 8(h)

Internal problem ID [4159]
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 : 8(h)
Date solved : Monday, January 27, 2025 at 08:41:00 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }-4 y^{\prime }+4 y&=2 x^{2}-4 x -1+2 x^{2} {\mathrm e}^{2 x}+5 x \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 42

dsolve(diff(y(x),x$3)-diff(y(x),x$2)-4*diff(y(x),x)+4*y(x)=2*x^2-4*x-1+2*x^2*exp(2*x)+5*x*exp(2*x)+exp(2*x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {{\mathrm e}^{-2 x} \left (\left (x^{3}+6 c_3 \right ) {\mathrm e}^{4 x}+3 x^{2} {\mathrm e}^{2 x}+6 c_{1} {\mathrm e}^{3 x}+6 c_{2} \right )}{6} \]

✓ Solution by Mathematica

Time used: 0.439 (sec). Leaf size: 44

DSolve[D[y[x],{x,3}]-D[y[x],{x,2}]-4*D[y[x],x]+4*y[x]==2*x^2-4*x-1+2*x^2*Exp[2*x]+5*x*Exp[2*x]+Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{6} \left (e^{2 x} x+3\right ) x^2+c_1 e^{-2 x}+c_2 e^x+c_3 e^{2 x} \]