12.19.18 problem section 9.3, problem 18

Internal problem ID [2165]
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 18
Date solved : Monday, January 27, 2025 at 05:42:58 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 87

dsolve(diff(y(x),x$4)-4*diff(y(x),x$3)+6*diff(y(x),x$2)-4*diff(y(x),x)+2*y(x)=exp(2*x)*(24+x+x^4),y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) c_1 +\sin \left (\frac {\sqrt {2}\, x}{2}\right ) c_2 \right )+\left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) c_3 +\sin \left (\frac {\sqrt {2}\, x}{2}\right ) c_4 \right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}+\frac {{\mathrm e}^{2 x} \left (x^{4}-8 x^{3}+12 x^{2}+49 x -62\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 102

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

\[ y(x)\to \frac {1}{2} e^{x-\frac {x}{\sqrt {2}}} \left (e^{\frac {x}{\sqrt {2}}+x} \left (x^4-8 x^3+12 x^2+49 x-62\right )+2 \left (c_4 e^{\sqrt {2} x}+c_2\right ) \cos \left (\frac {x}{\sqrt {2}}\right )+2 \left (c_1 e^{\sqrt {2} x}+c_3\right ) \sin \left (\frac {x}{\sqrt {2}}\right )\right ) \]