Internal problem ID [2290]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters
Method. page 556
Problem number: Problem 26.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+4 y^{\prime }-12 y-F \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 49
dsolve(diff(y(x),x$2)+4*diff(y(x),x)-12*y(x)=F(x),y(x), singsol=all)
\[ y \relax (x ) = c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{-6 x} c_{1}+\frac {\left (\left (\int F \relax (x ) {\mathrm e}^{-2 x}d x \right ) {\mathrm e}^{8 x}-\left (\int F \relax (x ) {\mathrm e}^{6 x}d x \right )\right ) {\mathrm e}^{-6 x}}{8} \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 63
DSolve[y''[x]+4*y'[x]-12*y[x]==F[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-6 x} \left (\int _1^x-\frac {1}{8} e^{6 K[1]} F(K[1])dK[1]+e^{8 x} \left (\int _1^x\frac {1}{8} e^{-2 K[2]} F(K[2])dK[2]+c_2\right )+c_1\right ) \\ \end{align*}