Zero everywhere, or nowhere zero. example: we know that y,(x) = cos x and y,(x) = sin x are solutions to y' +y = 0. since p o in this case, in light of abel's f

zero everywhere, or nowhere zero. Example: We know that y,(x) = cos x and y,(x) = sin x are solutions to y' +y = 0. Since p O in this case, in light of Abel's formula, we see VWronskian W(x) of y, and y, must be a constant. We confirm it by explicit computations: W(x) = cos x(sin x) - (cos x)' sin x = cos2 x + sin2 x = 1. Example: The functions y, (x) = ex and y,(x) = xe* are solutions to y" - 2y' + y = 0. Since p = ...(9) -2, we have W(x) = cex for some constant c. Explicit computation gives W(x) = e'(xe*) - (e*)'e* = e'(ex+ xe") - xe2x = ex, C = 1. .(10) sO