Question

Find the solution of the differential equation that satisfies the given initial condition.

du/dt = (2t + sec^2(t))/(2u), u(0) = -5

Answer 1

`\displaystyle (du)/(dt) = (2t + \sec^2 t)/(2u)\ \Rightarrow\ 2u\, du = \left( 2t + \sec^2 t \right) dt\ \Rightarrow\\ \\ \textstyle\int 2u\, du = \int \left( 2t + \sec^2 t \right) dt\ \Rightarrow\ u^2 = t^2 + \tan t + C, \\ \\ \text{where } \big[u(0\big]^2 = 0^2 + \tan 0 + C\ \Rightarrow\ C = (-5)^2 = 25. \\ \\ \text{Therefore, } u^2 = t^2 + \tan t + 25,\text{ so } u = \pm √(t^2 + \tan t + 25). \\ \\ \text{Since } u(0) = -5\text{, we must have }\ u = - √(t^2 + \tan t + 25).`