Use separation of variables to find the solution to the differential equation:
4 (du/dt) = u^2,
subject to the initial condition u(0)=6.
So far, I have:
4 du = u^2 dt
4/u^2 du = dt
-4/u = t+C
I am unsure what to do from this point...
oh come on.
0+C = -4/6
C = -2/3
-4/u = t - 4/3
u = -4/(t - 4/3) = 12/(4-3t)
To continue with the solution, we need to integrate both sides of the equation. Let's start with the left side:
∫(4/u^2) du = ∫dt.
The integral on the left side can be solved by using the power rule of integration. Integrating the left side, we get:
-4/u = t + C,
where C is the constant of integration.
To solve for u, we need to isolate u on one side of the equation. To do this, we can multiply both sides by u:
-4 = u(t + C).
Next, we can use the initial condition u(0) = 6 to determine the value of the constant C. Substituting t = 0 and u = 6 into the equation, we have:
-4 = 6(0 + C).
Simplifying,
-4 = 6C.
Divide both sides by 6:
C = -4/6 = -2/3.
Now we can substitute this value back into our equation to get the final solution:
-4/u = t - 2/3.
To solve for u, we can multiply both sides by -1 and rearrange the equation:
-1/(-4/u) = -1/(t - 2/3),
u/4 = -1/(t - 2/3),
u = -4/(t - 2/3).
Therefore, the solution to the given differential equation with the initial condition u(0) = 6 is:
u(t) = -4/(t - 2/3).
To continue solving the differential equation using separation of variables, we will proceed as follows:
From the equation you have obtained so far: -4/u = t + C
Step 1: Solve for u
To isolate u, we can start by multiplying both sides of the equation by -1/4, which gives us:
1/u = -1/4(t + C)
Then, we can reciprocate both sides to obtain:
u = -4/(t + C)
Step 2: Apply the initial condition
Now we can use the initial condition u(0) = 6 to determine the value of the constant C. Substituting t = 0 and u = 6 into the equation u = -4/(t + C), we have:
6 = -4/(0 + C)
Simplifying further, we get:
6 = -4/C
To solve for C, we can multiply both sides of the equation by C:
6C = -4
Then, divide both sides by 6:
C = -4/6
Simplifying further:
C = -2/3
Now that we have the value of C, we can write the final solution:
u = -4/(t - 2/3)
So, the solution to the given differential equation with the initial condition u(0) = 6 is u = -4/(t - 2/3).