MAXWELL'S EQUATIONS AND ELECTROMAGNETIC WAVES — Edward Purcell & David Morin
Physics — E&M · linear-combinations-of-solutions
Problem
Homogeneous linear differential equations have the property that the sum, or any linear combination, of two solutions is again a solution. (``Homogeneous'' means there's a zero on one side of the equation.) Consider, for example, the second-order equation (although the property holds for any order),
Ax¨+Bx˙+Cx=0.
Show that if x1(t) and x2(t) are solutions, then the sum x1(t)+x2(t) is also a solution. Show that this property does not hold for the nonlinear differential equation Ax¨+Bx˙2+Cx=0.