Solve the separable differential equation for $u$ \[ \frac{d u}{d t}=e^{5 u+6 t} \] Use the following initial condition: $u(0)=15$. \[ u= \]

Step 1 ：Separate the variables in the given differential equation. This gives us \(\frac{1}{e^{5u}} du = e^{6t} dt\).

Step 2 ：Integrate both sides with respect to their respective variables.

Step 3 ：Solve for \(u\) to get \(u = -\frac{1}{5}e^{-5u} = C + \frac{1}{6}e^{6t}\).

Step 4 ：Use the initial condition \(u(0)=15\) to find the constant of integration. This gives us \(C = -\frac{1}{6} - \frac{1}{5}e^{-75}\).

Step 5 ：Substitute \(C\) back into the equation for \(u\) to get \(u = \log\left(\left(\frac{6^{1/5}}{5e^{75}e^{6t} - 5e^{75} - 6}\right)^{1/5}e^{15}\right)\).

Step 6 ：\(\boxed{u = \log\left(\left(\frac{6^{1/5}}{5e^{75}e^{6t} - 5e^{75} - 6}\right)^{1/5}e^{15}\right)}\) is the final solution to the differential equation.