DIFFERENTIAL EQUATIONS verify that the indicated function yd

DIFFERENTIAL EQUATIONS:

verify that the indicated function y=d(x) is an explicit solution of the given first-order differential equation. ing simply as afunction, give its domain. Then by consid- ering as a solution of the differential equation, give at least one interval I of definition. by consider

Solution

y = 5 tan5x

y\' = 5 sec^2(5x)

25 + y^2 = 5 sec^2(5x)

25 + 25 tan^2(5x) = 5 sec^2(5x)

25(1+tan^2(5x)= 5 sec^2(5x)

25(1+tan^2(5x)) = 5 sec^2(5x)

5 sec^2(5x) = 5 sec^2(5x)

hence proved