a) verify that y=tan(x+c) is a one parameter family of solutions of the differen

Question

a) verify that y=tan(x+c) is a one parameter family of solutions of the differential equation y'=1+y^2.

b) since f(x,y)=1+y^2 and df/dy=2y are continuous everywhere, the region R in theorem 1.2.1 can be taken to be the entire xy-plane. use the family of solutions in part (a) to find an explicit solutions of the first order initial value problem y'=1+y^2, y(0)=0. even though x_0=0 is in the interval (-2,2), explain why the solution is not defined on this interval.

c) determine the largest interval I of definition for the solution of the initial value problem in part (b)

Explanation / Answer

y'=1+y^2.

dy/( 1+y^2.) = dx now integrate both side

we will get arctan(y) = x+C or Y = tan(x+C) for any integration constant value C.

as far as I under stood the problem the solution set would all the points on the tan(X) curve which is continuous for all value except at (n*pi/2) values where n is odd number becuse there it goes to infinity.

the lartgest interval will be (-n*pi/2, n*pi/2) for n is the odd number

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