Given the ODE Given y50 solve this is an exact equations pro

Given the O.D.E.:

Given y(5)=0, solve (this is an exact equations problem)

For what intervals of t and y are existence and uniqueness satisfied?

Solution

The only clue you give is that this is an exact equation.
So I\'m thinking you meant:

(2ty/(1+t²) 2) dt (2 ln(1+t²)) dy = 0
(2ty/(1+t²) 2) dt + (ln(1+t²) 2) dy = 0

Check that equation is indeed exact:
d/dy (2ty/(1+t²) 2) = 2t/(1+t²)
d/dt (ln(1+t²) 2) = 2t/(1+t²)

Solution is of the form: f(t,y) = C, where
f/t = 2ty/(1+t²) 2
f/y = ln(1+t²) 2

f(t,y) = ( 2ty/(1+t²) 2) dt = y ln(1+t²) 2t + g(y)
f(t,y) = (ln(1+t²) 2) dy = y ln(1+t²) 2y + h(t)

Comparing both value of f(t,y) we see that g(y) = 2y, h(t) = 2t
f(t,y) = y ln(1+t²) 2y 2t

Solution: y ln(1+t²) 2y 2t = C

Find C given y(5) = 0
0 0 10 = C
C = 10

y ln(1+t²) 2y 2t = 10

Solving for y, we get:

y (ln(1+t²) 2) = 2t10
y = 2 (t 5) / (ln(1+t²) 2)