Find the function satisfying the differential equation ftft5

Find the function satisfying the differential equation

f?(t)?f(t)=5t

and the condition f(3)=3 .
f(t)=

Solution

f(t) = y

y\' - y = 5t

IF = ( e^int p(t)dt)

IF = (e^int(-1 dt))

IF = e^(-t)

Solution:

y*IF = int (IF * q(t) dt)

y * e^(-t) = int (e^(-t) * 5t dt)

take -t = x

dt = -dx

y * e^(x) = int (e^x * -5x * -dx)

ye^x = int (e^x (5x) dx)

ye^x = 5 int (e^x (x-1 + 1) dx)

ye^x = 5 (e^x (x-1)) + c

ye^x = 5x*e^x - 5e^x + c

take x=-t

ye^(-t) = -5t*e^(-t) -5e^(-t) + c

y = -5t - 5 + ce^t

y(3) = 3

3 = -5*3 - 5 + ce^3

23 = ce^3

c = 23/e^3

y = -5t - 5 + ce^t

y = -5t -5 + 23e^(t-3)