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, crank it out:
4 du = u^2 dt
4/u^2 du = dt
-4/u = t+C
now plug in (0,6) to get C.
To solve the given differential equation using separation of variables, we need to separate the variables u and t on opposite sides of the equation:
4 (du/dt) = u^2
Divide both sides by u^2:
(4/ u^2) du/dt = 1
Now we can separate the variables by multiplying both sides by dt:
(4/ u^2) du = dt
Integrate both sides with respect to their respective variables:
∫ (4/ u^2) du = ∫ dt
To integrate the left side, use the power rule for integration:
4∫ (1/ u^2) du = ∫ dt
Let's solve the left side first:
4∫ (1/ u^2) du = 4(-1/u) + C1
Simplifying the left side of the equation:
-4/u + C1
Now, let's solve the right side:
∫ dt = t + C2
Combining the equations together:
-4/u + C1 = t + C2
Since we are given the initial condition u(0) = 6, we can substitute this value into the equation:
-4/6 + C1 = 0 + C2
Simplifying:
-2/3 + C1 = C2
Now we can combine the constants into a single constant:
C1 = C2 + 2/3
Substituting this back into the equation:
-4/u + C2 + 2/3 = t + C2
Simplifying:
-4/u + 2/3 = t
Finally, we can solve for u:
-4/u = t - 2/3
Multiply both sides by -u:
4 = -(t - 2/3)u
Divide both sides by (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 find the solution to the given differential equation using separation of variables, we need to separate the variables u and t on either side of the equation and integrate both sides separately.
Given the differential equation: 4(du/dt) = u^2.
Step 1: Rearrange the equation to separate the variables:
(1/u^2) du = (1/4) dt.
Step 2: Integrate both sides with respect to their respective variables:
∫(1/u^2) du = ∫(1/4) dt.
Step 3: Evaluate the integrals:
∫(1/u^2) du = -1/u + C1, where C1 is the constant of integration.
∫(1/4) dt = (1/4) t + C2, where C2 is the constant of integration.
Step 4: Combine the two evaluated integrals:
-1/u + C1 = (1/4) t + C2.
Step 5: Apply the initial condition u(0) = 6:
-1/6 + C1 = 1/4 * 0 + C2.
-1/6 + C1 = C2.
Step 6: Rearrange the equation to solve for C1:
C1 = C2 + 1/6.
Step 7: Substitute C1 back into the equation:
-1/u + C2 + 1/6 = (1/4) t + C2.
Step 8: Simplify the equation:
-1/u = (1/4) t + 1/6.
Step 9: Rearrange the equation to solve for u:
-1/u = (1/4) t + 1/6.
Step 10: Take the reciprocal of both sides:
u = -6/(t/4 + 1/6).
Therefore, the solution to the given differential equation with the initial condition u(0) = 6 is:
u(t) = -6/(t/4 + 1/6).