Verify that the indicated function y = f(x) is an explicit solution of the given first-order differential equation. Proceed as, by considering f simply as a function and give its domain. Then by considering f as a solution of the differential equation, give at least one interval I of definition. y = 25 + y2; y = 5 tan 5x

Hence the domain of the function y = 5 tan 5x is ,

and largest interval of definition for the solution is

Step-by-step explanation:

Considering the differential equation

and the function  y = 5 tan 5x

Find the domain of the function  y = 5 tan 5x

Since then determine the zeros of the denominator, cos 5x

The zeros of the denominator of cos 5x is

Hence the function  y = 5 tan 5x exist on

Therefore, the domain of the function y = 5 tan 5x is  

Differential the function  y = 5 tan 5x with respect to x

Use  trigonometry identity ,

Substitute

Therefore the function y = 5 tan 5x satisfies the differential equation

Now , Find the largest interval of the solution

Since the tangent function is periodic with period π so take interval

since , the function y = 5 tan 5x is not differentiatable  at

Hence the domain of the function y = 5 tan 5x is ,

and largest interval of definition for the solution is

just needed some points but try socratic