Find u Solution Rewrite e5u 10t as two terms e5u 10t e5u

Solution

Rewrite e^{5u + 10t} as two terms: e^{5u + 10t} = e^(5u)e^(10t) = du/dt du / e^(5u) = [e^(-5u)]du = [e^(10t)]dt The rule is: Integral e^(cx) = (1/c)e^(cx) Integral: [e^(-5u)]du ----> (-1/5)e^(-5u) Integral: [e^(10t)]dt --------> (1/10)e^(10t) Add these and a constant and do some manipulation( C will change but C is just a constant so I will simply keep calling it C): (-1/5)e^(-5u) = (1/10)e^(10t) + C e^(-5u) = C - (5/10)e^(10t) Take ln of both sides: -5u = ln[C - (5/10)e^(10t)] u = (-1/5)ln[C - (5/10)e^(10t)] This seems a rather odd answer but I can see nothing wrong. And then use u(0) = 6 to solve for C 6 = (-1/5)ln[C - 5/10] e^(-50) = C - 5/10 C = e^(-50) + 5/10 Final equation is: u = (-1/5)ln[e^(-50) + 5/10 - (5/10)e^(10t)]