What is y as a function of t

Question

I need help with this problem

 

Suppose that y′=ky, y(0)=12, and y′(0)=13. What is y as a function of t ?

Answer 1

Since we know both y and y' at the same instant of time t=0, we can solve for k using the equation y'=ky.  The function y(t) can be derived by changing y' into dy/dt.  

 

Solve for k:

y'=ky

13=k*12
k=13/12

 

Partially derive y(t):

y'=ky

dy/dt=ky

(1/y)dy=kdt

int((1/y)dy=int(kdt)

ln(y)=kt+c

y=exp(kt+c)

 

Solve for c:

y=exp(kt+c)

12=exp(k*0+c)

12=exp(c)

c=ln(12)

 

Fully derive y(t):

y=exp((13/12)t+ln(12))

y=12exp((13/12)*t)

y'=13exp((13/12)*t)

 

y and y' can be checked by substitution.