Internal problem ID [1585]
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 35.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-2 y-F \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 63
dsolve(diff(y(x),x$3)+2*diff(y(x),x$2)-diff(y(x),x)-2*y(x)=F(x),y(x), singsol=all)
\[ y \relax (x ) = \left (\int \frac {{\mathrm e}^{-x} F \relax (x )}{6}d x \right ) {\mathrm e}^{x}+\left (\int \frac {F \relax (x ) {\mathrm e}^{2 x}}{3}d x \right ) {\mathrm e}^{-2 x}-\left (\int \frac {{\mathrm e}^{x} F \relax (x )}{2}d x \right ) {\mathrm e}^{-x}+{\mathrm e}^{x} c_{1}+c_{2} {\mathrm e}^{-2 x}+c_{3} {\mathrm e}^{-x} \]
✓ Solution by Mathematica
Time used: 0.054 (sec). Leaf size: 88
DSolve[y'''[x]+2*y''[x]-y'[x]-2*y[x]==f[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-2 x} \left (\int _1^x\frac {1}{3} e^{2 K[1]} f(K[1])dK[1]+e^x \left (\int _1^x-\frac {1}{2} 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 )+c_1\right ) \\ \end{align*}