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Solve the separable differential equation for u dudt=e6u+3t. Use the following initial condition: u(0)=−5 . u=

User BBog
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The solution to the differential equation with the given initial condition is:
e^((6u)) = 2 ln|t| - 5/6

To solve the separable differential equation: u
du/dt = e^((6u)) + 3t, we can rearrange the equation as follows:


du/(e^((6u))) = (1/3t) dt

Now, we can integrate both sides.

On the left side, we can use the substitution v = 6u, which means dv = 6 du:

(1/6) ∫
e^vdv = (1/3) ∫ (1/t) dt

Integrating both sides, we get:

(1/6)
e^v = (1/3) ln|t| + C

Now, substitute back v = 6u:


(1/6) e^((6u)) = (1/3) ln|t| + C

Finally, to find the particular solution, we can use the given initial condition u(0) = -5:


(1/6) e^((6(-5))) = (1/3) ln|0| + C

Simplifying, we find C = -5/6.

Therefore, the solution to the differential equation with the given initial condition is:


e^((6u)) = 2 ln|t| - 5/6