Solve the separable differential equation for u, du/dt=e^(4u+7t).
Use the following initial condition: u(0)=2
du/dt = e^(4u+7t)
du/dt = e^(4u) * e^(7t)
e^(-4u) du = e^(7t) dt
-1/4 e^u = 1/7 e^(7t) + C
e^u = 4C - 4/7 e^(7t)
and the point (0,2) yields e^2 = 4C - 4/7
You can massage that in various ways, but i can't just now see how wolframalpha.com finishes up with
u = -1/4 ln(4/7 - 1/e^8 - 4/7 e^(7t))