Separable Differential Equation explained So i have the equation dy/dt = 1+y looking at the answers my lecture has given it states d/dt(1+y) = 1 + y then 1+y(t)=Ae^t where A is a constant Can someone explain these steps please?

Hanna Webster

Hanna Webster

Answered question

2022-11-18

Separable Differential Equation explained
So i have the equation d y d t = 1 + y
looking at the answers my lecture has given it states
d d t ( 1 + y ) = 1 + y
then 1 + y ( t ) = A e t where A is a constant
Can someone explain these steps please?

Answer & Explanation

erlent00s

erlent00s

Beginner2022-11-19Added 15 answers

Since 1 is a constant,
d d t ( 1 + y ) = d d t ( 1 ) + d d t ( y ) = 0 + d y d t = d y d t   .
That's why your original DE can be written as
d d t ( 1 + y ) = 1 + y   .
Now this means that 1 + y is a function which is equal to its own derivative. What function could that be? Basically it must be e t , a function which equals its own derivative, but if you think about you will see that the same is true for any constant times e t . These are the only possibilities, so the solution is
1 + y = A e t   .
is a neat trick, but might be confusing if you have not seen a lot of DEs. If that is the case then you might do better to use the standard method:
d y d t = 1 + y 1 1 + y d y d t = 1 1 1 + y d y d t d t = 1 d t 1 1 + y d y = 1 d t ln ( 1 + y ) = t + C 1 + y = e t + C = e C e t 1 + y = A e t   .

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?