Internal problem ID [1587]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 9 Introduction to Linear Higher Order Equations. Section 9.4. Variation of
Parameters for Higher Order Equations. Page 503
Problem number: section 9.4, problem 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 }-5 y^{\prime \prime }+4 y-F \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 85
dsolve(diff(y(x),x$4)-5*diff(y(x),x$2)+4*y(x)=F(x),y(x), singsol=all)
\[ y \relax (x ) = -\left (\int \frac {{\mathrm e}^{-x} F \relax (x )}{6}d x \right ) {\mathrm e}^{x}-\left (\int \frac {F \relax (x ) {\mathrm e}^{2 x}}{12}d x \right ) {\mathrm e}^{-2 x}+\left (\int \frac {{\mathrm e}^{x} F \relax (x )}{6}d x \right ) {\mathrm e}^{-x}+\left (\int \frac {F \relax (x ) {\mathrm e}^{-2 x}}{12}d x \right ) {\mathrm e}^{2 x}+c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{-2 x}+c_{3} {\mathrm e}^{-x}+c_{4} {\mathrm e}^{2 x} \]
✓ Solution by Mathematica
Time used: 0.047 (sec). Leaf size: 117
DSolve[y''''[x]-5*y''[x]+4*y[x]==f[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-2 x} \left (\int _1^x-\frac {1}{12} e^{2 K[1]} f(K[1])dK[1]+e^x \left (\int _1^x\frac {1}{6} e^{K[2]} f(K[2])dK[2]+c_2\right )+e^{3 x} \left (\int _1^x-\frac {1}{6} e^{-K[3]} f(K[3])dK[3]+c_3\right )+e^{4 x} \left (\int _1^x\frac {1}{12} e^{-2 K[4]} f(K[4])dK[4]+c_4\right )+c_1\right ) \\ \end{align*}