Why do we use initial conditions at the end of problems e.g in solving for differential equations

Question

Normally when we solve differential equations, we use initial conditions to find the values of the integration constants we find, at the end. When solving for a variable, would it be wrong to apply the initial conditions in the middle of a problem, as opposed to applying it at the end?

Answer 1

I'm not entirely sure what you mean by "insert an initial condition midway through a problem", but I'll give it a shot.

Remember that an initial condition just means what's happening at some starting time. Your differential equation is describing what's happening all the time. If you insert your condition too early, you're throwing away most of that information.

To take a simple example, let's solve the initial value problem $y' = ty$, $y(0) = 2$. If we insert $t = 0$, $y = 2$ immediately, we get $\frac{d}{dt}2 = 0 \cdot 2$, which is $0 = 0$; now there's nothing to solve, which is great, but we also don't have a solution. The original equation $y' = ty$ was saying something about all values of $t$; by inserting $t = 0$, we were saying "no, I only care about $t = 0$", which isn't true.

If we instead start solving, we get:

$\frac{dy}{dt} = ty$

$dy = tydt$

$\frac{1}{y}dy = tdt$

We still don't want to insert values here - the "$dy$" and the "$dt$" are trying to talk about what happens as $y$ and $t$ change. If we insert our initial condition, we're requiring them not to change. Integrating both sides:

$\ln|y| = \frac{1}{2}t^2 + c$

Now, we could substitute our initial condition here, to get $\ln|2| = \frac{1}{2}0^2 + c$, so $c = \ln{2}$. That would work great. There isn't much point, though, because we have to finish solving the equation anyway.

$|y| = e^{\frac{1}{2}t^2 + c}$

$y = \pm e^{\frac{1}{2}t^2 + c}$

Now, a common step here is to move the integration constant out - it's just an arbitrary constant, so $\pm e^c$ is also just an arbitrary constant. That way we have the much simpler $y = ce^{\frac{1}{2}t^2}$. We couldn't have done it that way if we'd solved for $c$ earlier, because then $c$ wouldn't be arbitrary!

Now that we have $y = ce^{\frac{1}{2}t^2}$, we can plug in our initial condition safely and get $2 = c$. So our solution is $y = 2e^{\frac{1}{2}t^2}$.

Moral of the story: We can plug in the initial condition a little early; basically, as soon as we're done dealing with derivatives. But it isn't very convenient, so we usually just do it at the end.

Answer 2

Excellent idea.

In fact there are methods such as Laplace Transform methods for which you apply the initial condition at the beginning and find the solution based on the given data later.

That saves time towards the end because the constants are already found.

As you know when you solve linear differential equations your particular solution does not depend on initial conditions.

You have to find the particular solution first and add it to the homogeneous solution and finally find the constants.

Thus for some methods you have to wait until you have found the general solution and find the constants afterward.