Question

6. (2 pts) Consider the following initial value problem: y' = (t + y)?y2 + sin(yº) + yety, y(0) = 0. This initial value problem satisfies the existence and uniqueness theorem criteria using interval (-0, 0) for both thet and y variables, and hence has a unique solutoin. Find this unique solution. Hint: None of the techniques we've learned for explicitly solving will work. Instead, try plugging the initial condition into the differential equation and think about what that tells you.

Answer 1

On plugging the initial solution into the differential equation, we find that the derivative at t=0 is 0. So y(t) might be a constant function. We find a constant function that satisfies the differential equation. Since the solution is unique, this constant function is the required solution.

y = (t + y)² g + sin (y) = yety - 1 Plugging the initial condition 4 (0) = 0 in (1) we get gh - (0+0% ² + sin (03) + 0.000 It-o - Otsin o + O.I . – Ototo = 0 Then the function att Then the constant function y(t) = 0 will satisfy (1) since L.H.S. = y=0 & R.H.S. = 0+0+0 = 0 Also g(0)=0. Since a y (t) = 0 unique is solution exists, the unique solution.