how would i solve the differential equation
dx/dt=-kx^2
i.e. solve for x as a function of t.
I've been told to rearrange the equation to yield
dx/x^2=-k dt
but what can and should I do now?
integral of x^n dx = (1/(n+1)) x^(n+1) in general
integral of x^-2 dx = (1/-1) x^-1
which is( 1/x ) + constant of integration
the right side is -k t + constant of integration
1/x = -k t + c
x = 1/ (-kt+c)
undefined when kt = c
Well, well, well, looks like you found yourself a tricky little differential equation there! Don't worry, Clown Bot is here to make it fun.
Now that you have the equation dx/x^2 = -k dt, you can start by integrating both sides. The integral of dx/x^2 might seem a bit intimidating, but fear not! Let's break it down.
If we rewrite the equation as x^(-2) dx = -k dt, we can integrate both sides. So, on the left side, we integrate the function x^(-2) with respect to x, and on the right side, we integrate -k with respect to t.
Integrating x^(-2) with respect to x might bring up some memories of high school math. Well, let Clown Bot refresh your memory. The integral is -x^(-1), or -1/x. Remember, we also need to add a constant of integration, let's call it C1.
So now our equation becomes -1/x = -kt + C1. Don't worry, we're almost there!
To get rid of that pesky negative sign, we can multiply both sides of the equation by -1. Yeah, we can't have negativity ruining our day. That gives us 1/x = kt - C1. Almost done!
Finally, to solve for x, we can take the reciprocal of both sides. That's right, we're flipping the equation around, just like a circus acrobat flips through the air. So, we end up with x = 1/(kt - C1).
And there you have it! The solution to your differential equation, x as a function of t, is x = 1/(kt - C1).
But let's not forget about our friend C1! It represents the constant of integration, so you'll need some initial condition or boundary condition to determine its value. So don't forget to clown around with that too!
Hope you enjoyed the circus act! If you have any more puzzling questions, Clown Bot is always here to add some laughter to the learning process.
To solve the differential equation dx/dt = -kx^2, you can rearrange the equation as you've been told:
dx/x^2 = -k dt.
Now, we can integrate both sides of the equation:
∫dx/x^2 = ∫-k dt.
The integral of dx/x^2 can be written as:
∫(x^(-2)) dx = ∫-k dt.
Now, let's integrate each side separately:
∫(x^(-2)) dx = -k ∫dt.
To integrate x^(-2) with respect to x, we can use the power rule of integration. The integral will be:
-1/x = -kt + C,
where C is the constant of integration.
Now, we need to solve for x. To do this, we can take the reciprocal of both sides of the equation:
1/x = kt - C.
Finally, let's take the reciprocal one more time:
x = 1/(kt - C).
Therefore, the solution to the differential equation dx/dt = -kx^2 is x = 1/(kt - C), where k and C are constants.
To solve the given differential equation dx/dt = -kx^2, you can rearrange the equation as dx/x^2 = -k dt, as you were told. This rearrangement allows us to separate the variables and integrate both sides of the equation.
To proceed, let's integrate both sides of the equation:
∫ (dx/x^2) = ∫ (-k dt)
The integral of dx/x^2 is a standard integral and can be evaluated as:
-1/x = -kt + C
Here, C is the constant of integration, which is determined by initial conditions or additional information given in the problem.
To solve for x, we can take the reciprocal of both sides of the equation:
x = -1/(-kt + C)
Simplifying further, we get:
x = 1/(kt - C)
So, the solution to the differential equation dx/dt = -kx^2 is x = 1/(kt - C), where C is the constant of integration determined by additional information or initial conditions.