Find the function that satisfies

Question

dJ/dt=4J with J(0)= 5

Answer 1

Here's another shot at it (!) :

 

dJ/dt = 4J,   so dJ/J  = 4dt

 

∫dJ/J = 4∫dt,  so ln[J] = 4t + C

 

Therefore: J = e^{(4t + C)} = C'e^4t

 

5 = C'e^4*0  =  C'*1, so C' = 5

 

So: J = 5e^4t

 

 

Answer 2

Noting that the exponential function has the property that it is  proportional to it's derivative so if we let j=ce^kt then dj/dt=cke^kt substituting in the original equation gives

 

cke^kt=4ce^kt so if k=4 then j=ce^4t satisfies the differential equation to find  c we use the initial condition j(0)=5 i,e, 5=c

 

so the required function is j(t)=5e^4t

 

Hope this helps

 

Jim

Answer 3

Find the function J(t) that satisfies dJ(t)/dt = 4J(t)

 

      J(t) = ∫4J(t) dt

 

      J(t) = e^4t + C

 

where C is a constant.

 

      J(t=0) = 5 = e^4*0 + C = 1 + C

      4 = C

 

      f(J) = e^4t + 4

 

Check:

 

      dJ(t)/dt = d(e^4t)/dt*d(4t)/dt + d(4)/dt = 4e^4t = 4J(t)