Hiya,
Dt/dr =k/r^2
Is this directly solvable or do I need to separate variables.
I've got t=-k/r+c from the former and something along the lines of r^3=kt+c from the latter.
Think I'm doing it all wrong.
Thanks
dt/dr = k/r^2
dt = k dr/r^2
try t = k (-1/r) + c
I do not know where you got the other.
Thanks, just testing out methods and got carried away!
Hi there,
To solve the differential equation dt/dr = k/r^2, you can indeed separate variables. Here's the step-by-step process:
Step 1: Move the variables around to separate dt and dr: dt = k/r^2 * dr.
Step 2: Integrate both sides of the equation. For the left side, you integrate with respect to t, and for the right side, you integrate with respect to r:
∫dt = ∫k/r^2 * dr.
Step 3: Evaluate the integrals:
On the left side, integrating dt gives you t. On the right side, you have ∫k/r^2 * dr, which you can rewrite as k∫r^(-2) * dr.
Step 4: After integrating, you'll get:
t = -k/r + c.
Now let's verify if the expression you provided, t = -k/r + c, is correct.
To do this, take the derivative of -k/r + c with respect to r:
d/dt(-k/r + c) = k/r^2.
As you can see, the derivative matches the original differential equation dt/dr = k/r^2. Therefore, the expression t = -k/r + c is indeed the solution to the given differential equation.
It seems you got it right! Keep in mind that the expression r^3 = kt + c you mentioned is not a solution to the given differential equation, dt/dr = k/r^2.
I hope this clarifies things for you. Let me know if you have any further questions!