How to solve a separable differential equation
that's of this differential equation we have dx/dt it's equal to t over x times e to the T plus 2x
power how is this separable yes yes because if you look at T plus 2x they are in the exponent
right we can use one of the group exponent which is separated if you look at this right here as
T on the top over X right here when you add the exponent it's a same as multiply each of this e
to that right so I can look at this as e to the T times e to the 2x so let me write this down right
here we have DX DT it's equal to let me just put down all the eggs together first okay so
we will have 1 over X right here and then e2 it works together like this and this right here is
a function of X and then we multiplied by T on top over e to the T on the side right here so
you see it is separable and now you see here we have DX over DT let's multiply did here on both
sides so that they can so and I want to bring all the eggs together to the other side right
so let's multiply X e to the 2x so they cancel I'll have to do it right here as well it's 2012
and this right here are good because all the T's are together so all together right here will have
X e to the 2x and this is yet and this is equal to to integrate he over each would he have to do
cat is this T times e to the negative he okay because I recognize I had to do integration by
parts right here I also have to do inter question by parts right here so integrate this integrator
and the best way to do integration by part is of course to use the t-i method to a TI method
right here real quick plus minus unless I I'm seeing a nice tree of them so let's put the T
right here as the X and then I that's integrate e xx 2x two differential X we care 1 and then 0
and then to integrate this we have 1/2 e to the 2x and then integrate the again we have 1 over
4 e to the 2x right so I have to integrate this running through the DI method with the
integration by parts right this this it's pretty much the answer don't forget the sign on the side
plus minus and when you do it the hacker knows they are the answers already so we will have x
times this which is 1/2 X and then e to the 2x next what this time stamp which is minus
1 over 4 e to the 2x this is it for that right and I'm not going to put on plus a constant here
because I just have to do it on the right hand side let's focus on integrating this right here
and once again we need to use the TI method for the integration by parts and these two of them
three rows that's differentiate T so we kept one a zero that's integrate e to the negative T right
here we will have negative e to the negative T do again negative negative becomes positive so
we will just have e to the negative part lead right here diagonals are the answer this is 0
so it's great bring down the equal sign first we have this time step which is negative T e to the
negative T and this times that negative 1 times tab which is negative e to the negative T power
ok so that's the integral for that this is an integral result of that and I will just put on
plus C right here X is the dependent variable and he was the independent variable this nearly
no way to isolate the X right so we will leave the answer right here ok this is it and when
you cannot solve for the dependent variable this right here it's called the implicit solution ok
sometimes if you can solve for X we working softer Y if you can solve for the dependent
variable that's called the explicit solution but this is called a solution and this is it