Let ft be the solution of yy2ty y01 Which of the following i

Solution

Add y(t) to both sides:

( dy(t))/( dt)+y(t) = t y(t)^2

Divide both sides by -y(t)^2:

-(( dy(t))/( dt))/y(t)^2-1/(y(t)) = -t

Let v(t) = 1/(y(t)), which gives ( dv(t))/( dt) = -(( dy(t))/( dt))/y(t)^2:

( dv(t))/( dt)-v(t) = -t

Let mu(t) = exp( integral -1 dt) = e^(-t).
Multiply both sides by mu(t):

e^(-t) ( dv(t))/( dt)-e^(-t) v(t) = -e^(-t) t

Substitute -e^(-t) = ( d)/( dt)(e^(-t)):

e^(-t) ( dv(t))/( dt)+( d)/( dt)(e^(-t)) v(t) = -e^(-t) t

Apply the reverse product rule f ( dg)/( dt)+( df)/( dt) g = ( d)/( dt)(f g) to the left-hand side:

( d)/( dt)(e^(-t) v(t)) = -e^(-t) t

Integrate both sides with respect to t:

( d)/( dt)(e^(-t) v(t)) dt = -e^(-t) t dt

Evaluate the integrals:

e^(-t) v(t) = -e^(-t) (-t-1)+c, where c is an arbitrary constant.

Divide both sides by mu(t) = e^(-t):

v(t) = t+c e^t+1

Solve for y(t):

y(t) = 1/(v(t)) = 1/(t+ce^t+1)

For this function if we find derivative, only option D is correct