Solve the seperable differential equation for U.
du/dt = e^(5u+7t) - Using the following initial condition U(0) = 12
du/dt = e^5u * e^7t
du/e^5u = e^7t dt
e^-5u du = e^7t dt
(1/-5) e^-5u = (1/7) e^7t + C
7 e^-5u = -5 e^7t + c
when t = 0, u = 12
7 e^-60 = -5 + c
c = 5 +7/e^60 or c is about 5
7 e^-5u + 5 e^7t = 5
Solve for u
-29 + -4 + 5u
To solve the separable differential equation du/dt = e^(5u+7t) with the initial condition U(0) = 12, follow these steps:
Step 1: Separate the variables.
Write the equation as:
du / (e^(5u)) = e^(7t) dt
Step 2: Integrate both sides with respect to their respective variables.
Integrate the left side with respect to u:
∫(1 / e^(5u)) du = ∫1 du / e^(5u)
This integral can be simplified by using a substitution. Let's set v = 5u, then du = (1/5) dv. The integral becomes:
(1/5) ∫e^(-v) dv = (1/5) (-e^(-v)) + C1
Integrate the right side with respect to t:
∫e^(7t) dt = (1/7) e^(7t) + C2
Step 3: Combine the results and solve for U.
Now we have:
(1/5) (-e^(-v)) + C1 = (1/7) e^(7t) + C2
Let's simplify the constants by combining C1 and C2 into a single constant C:
(1/5) (-e^(-v)) + C = (1/7) e^(7t)
Step 4: Solve for U.
Since v = 5u, we can substitute 5u back into the equation:
(1/5) (-e^(-5u)) + C = (1/7) e^(7t)
To isolate u, rearrange the equation:
(1/5) (-e^(-5u)) = (1/7) e^(7t) - C
Multiply both sides by -5:
e^(-5u) = -5/7 e^(7t) + 5C
Take the natural logarithm (ln) of both sides:
-5u = ln(-5/7 e^(7t) + 5C)
Divide both sides by -5:
u = -ln(-5/7 e^(7t) + 5C) / 5
Finally, substitute the initial condition U(0) = 12 into the equation to find the specific solution value for C. Since U(0) = 12, we have:
12 = -ln(-5/7 e^(0) + 5C) / 5
Solve the equation for C, and then substitute the value of C back into the solution equation for u, and you will get the solution for u.
To solve the separable differential equation du/dt = e^(5u+7t), first separate the variables by moving all terms involving u to one side and terms involving t to the other side:
du / e^(5u) = e^(7t) dt
Next, integrate both sides of the equation with respect to their corresponding variables. The integral of du / e^(5u) is given by:
∫ (1 / e^(5u)) du = ∫ e^(7t) dt
To calculate these integrals, let's use the substitution method. Let y = 5u, then dy = 5du.
Substituting into the left-hand side of the equation:
∫ (1 / e^y) (1/5) dy = ∫ e^(7t) dt
Simplifying the left-hand side of the equation:
(1/5) ∫ e^(-y) dy = ∫ e^(7t) dt
To integrate e^(-y), recall that the integral of e^x is just e^x:
(1/5) ∫ e^(-y) dy = (1/5) e^(-y) = (1/5) e^(-5u)
Now we have:
(1/5) e^(-5u) = ∫ e^(7t) dt
Integrate the right-hand side of the equation:
(1/5) e^(-5u) = (1/7) e^(7t) + C
where C is the constant of integration.
To find the specific solution, substitute the initial condition U(0) = 12 into the equation. When t = 0, the right-hand side becomes:
(1/7) e^(7*0) + C = (1/7) e^0 + C = (1/7) + C
So, when t = 0, U(0) = 12 gives:
(1/5) e^(-5 * 12) = (1/7) + C
Now, you can solve for the constant C by rearranging the equation:
(1/5) e^(-60) - (1/7) = C
Using a calculator to evaluate e^(-60), you can substitute that value into the equation:
C ≈ 3.8234 - (1/7) ≈ 3.6894
Therefore, the final solution to the differential equation du / dt = e^(5u+7t) with the initial condition U(0) = 12 is:
(1/5) e^(-5u) = (1/7) e^(7t) + 3.6894