Differential Equations

Question

Find the solution to the differential equation dz/dt=9te^5z that passes through the origin. Could someone help me understand this?

Answer 1

Separate variables:  e^-5z dz = 9tdt

 

Integrate both sides:  (-1/5)e^-5z = 4.5t²+C

 

Since z = 0 when t = 0, -1/5 = 0+C.  So, C = -1/5

 

Therefore, (-1/5)e^-5z = 4.5t² - 1/5

 

                       e^-5z = 1 - 22.5t²

 

                        -5z = ln(1-22.5t²)

 

                           z = -0.2[ln(1-22.5t²)]