Internal problem ID [2277]
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 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-2 m y^{\prime }+m^{2} y-\frac {{\mathrm e}^{m x}}{x^{2}+1}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 36
dsolve(diff(y(x),x$2)-2*m*diff(y(x),x)+m^2*y(x)=exp(m*x)/(1+x^2),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{m x} c_{2} +{\mathrm e}^{m x} x c_{1} +{\mathrm e}^{m x} \left (-\frac {\ln \left (x^{2}+1\right )}{2}+x \arctan \left (x \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 35
DSolve[y''[x]-2*m*y'[x]+m^2*y[x]==Exp[m*x]/(1+x^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{m x} \left (-\log \left (x^2+1\right )+2 (x (\arctan (x)+c_2)+c_1)\right ) \\ \end{align*}