12.19.56 problem section 9.3, problem 56

Internal problem ID [2203]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined Coefficients for Higher Order Equations. Page 495
Problem number : section 9.3, problem 56
Date solved : Monday, January 27, 2025 at 05:43:30 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-3 y^{\prime \prime }-4 y^{\prime }+4 y&=2 \left (1+x \right ) {\mathrm e}^{x}+{\mathrm e}^{-2 x} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 50

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

\[ y = \frac {{\mathrm e}^{-2 x} \left (\left (x^{3}+x^{2}+\left (27 c_3 -2\right ) x +27 c_1 +\frac {10}{9}\right ) {\mathrm e}^{3 x}+\frac {3 x^{2}}{2}+\left (27 c_4 +2\right ) x +27 c_2 +1\right )}{27} \]

✓ Solution by Mathematica

Time used: 0.267 (sec). Leaf size: 66

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

\[ y(x)\to \frac {1}{54} e^{-2 x} \left (3 x^2+(4+54 c_2) x+2+54 c_1\right )+\frac {1}{243} e^x \left (9 x^3+9 x^2+9 (-2+27 c_4) x+10+243 c_3\right ) \]