28.2.19 problem 19

Internal problem ID [4462]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 19
Date solved : Monday, January 27, 2025 at 09:19:03 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-2 y^{\prime \prime }+4 y^{\prime }-8 y&={\mathrm e}^{2 x} \sin \left (2 x \right )+2 x^{2} \end{align*}

✓ Solution by Maple

Time used: 0.530 (sec). Leaf size: 58

dsolve(diff(y(x),x$3)-2*diff(y(x),x$2)+4*diff(y(x),x)-8*y(x)=exp(2*x)*sin(2*x)+2*x^2,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (80 c_{2} -2 \cos \left (2 x \right )-4 \sin \left (2 x \right )-5\right ) {\mathrm e}^{2 x}}{80}+\frac {\left (80 c_{1} -5\right ) \cos \left (2 x \right )}{80}+\frac {\left (80 c_3 +5\right ) \sin \left (2 x \right )}{80}-\frac {x^{2}}{4}-\frac {x}{4} \]

✓ Solution by Mathematica

Time used: 0.249 (sec). Leaf size: 61

DSolve[D[y[x],{x,3}]-2*D[y[x],{x,2}]+4*D[y[x],x]-8*y[x]==Exp[2*x]*Sin[2*x]+2*x^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{80} \left (-20 x (x+1)+5 (-1+16 c_3) e^{2 x}-2 \left (e^{2 x}-40 c_1\right ) \cos (2 x)-4 \left (e^{2 x}-20 c_2\right ) \sin (2 x)\right ) \]