Find the solution to the differential equation dzdt 6te5t t

Find the solution to the differential equation dz/dt = 6te^5t that passes through the origin. z =

Solution

Applying variable seperable method we have

dz / dt = 6t e^5z can be written as : dz / e^5z = 6t dt

: e^-5z  dz = 6t dt integrating on either side

- e^-5z / 5 = 3t² + c where C is the constant of integration

put t = 0 and z = 0 - 1 / 5 = c . substitute back in the above equation

- e^-5z  = 15 t² - 1 or e^-5z  = 1- 15 t² or - 5z = log ( 1 - 15 t²)

z = - 1 /5 log ( 1 - 15t²)