Internal problem ID [5113]
Book: THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T.
CHAU, CRC Press. Boca Raton, FL. 2018
Section: Chapter 3. Ordinary Differential Equations. Section 3.5 HIGHER ORDER ODE. Page
181
Problem number: Example 3.39.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+3 y^{\prime \prime }-4 y^{\prime }-4 y-f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 91
dsolve(diff(y(x),x$4)+4*diff(y(x),x$3)+3*diff(y(x),x$2)-4*diff(y(x),x)-4*y(x)=f(x),y(x), singsol=all)
\[ y \relax (x ) = \left (\int \frac {f \relax (x ) {\mathrm e}^{-x}}{18}d x \right ) {\mathrm e}^{x}+\left (\int -\frac {f \relax (x ) \left (3 x -4\right ) {\mathrm e}^{2 x}}{9}d x \right ) {\mathrm e}^{-2 x}-\left (\int \frac {f \relax (x ) {\mathrm e}^{x}}{2}d x \right ) {\mathrm e}^{-x}+\left (\int \frac {f \relax (x ) {\mathrm e}^{2 x}}{3}d x \right ) {\mathrm e}^{-2 x} x +c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{-2 x}+c_{3} {\mathrm e}^{-x}+c_{4} {\mathrm e}^{-2 x} x \]
✓ Solution by Mathematica
Time used: 0.07 (sec). Leaf size: 120
DSolve[y''''[x]+4*y'''[x]+3*y''[x]-4*y'[x]-4*y[x]==f[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-2 x} \left (x \int _1^x\frac {1}{3} e^{2 K[2]} f(K[2])dK[2]+\int _1^x-\frac {1}{9} e^{2 K[1]} f(K[1]) (3 K[1]-4)dK[1]+e^x \left (\int _1^x-\frac {1}{2} e^{K[3]} f(K[3])dK[3]+c_3\right )+e^{3 x} \left (\int _1^x\frac {1}{18} e^{-K[4]} f(K[4])dK[4]+c_4\right )+c_2 x+c_1\right ) \\ \end{align*}