12.20.16 problem section 9.4, problem 39

Internal problem ID [2237]
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
Date solved : Monday, January 27, 2025 at 05:43:56 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y&=F \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 84

dsolve(diff(y(x),x$4)-5*diff(y(x),x$2)+4*y(x)=F(x),y(x), singsol=all)
 

\[ y = -\frac {\left (\left (\int {\mathrm e}^{-x} F \left (x \right )d x \right ) {\mathrm e}^{3 x}-6 c_1 \,{\mathrm e}^{3 x}-\frac {\left (\int F \left (x \right ) {\mathrm e}^{-2 x}d x \right ) {\mathrm e}^{4 x}}{2}-6 c_4 \,{\mathrm e}^{4 x}-\left (\int {\mathrm e}^{x} F \left (x \right )d x \right ) {\mathrm e}^{x}-6 c_3 \,{\mathrm e}^{x}+\frac {\left (\int F \left (x \right ) {\mathrm e}^{2 x}d x \right )}{2}-6 c_2 \right ) {\mathrm e}^{-2 x}}{6} \]

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 130

DSolve[D[y[x],{x,4}]-5*D[y[x],{x,2}]+4*y[x]==f[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-2 x} \left (\int _1^x-\frac {1}{12} e^{2 K[1]} f(K[1])dK[1]+e^x \int _1^x\frac {1}{6} e^{K[2]} f(K[2])dK[2]+e^{3 x} \int _1^x-\frac {1}{6} e^{-K[3]} f(K[3])dK[3]+e^{4 x} \int _1^x\frac {1}{12} e^{-2 K[4]} f(K[4])dK[4]+c_2 e^x+c_3 e^{3 x}+c_4 e^{4 x}+c_1\right ) \]