\( t^{-1} \frac{d y}{d t}=9 \sin ^{2} y, y(1)=\frac{3 \pi}{4} \) The solution is \( \square \). (Type an implicit solution. Type an equation using \( t \) and \( y \) as the variables.)

To solve the differential equation \[ t^{-1} \frac{d y}{d t} = 9 \sin^2 y, \] we can first rearrange it into a more standard form. Multiplying both sides by \( t \) gives: \[ \frac{d y}{d t} = 9 t \sin^2 y. \] This is a separable differential equation, so we can separate the variables \( y \) and \( t \): \[ \frac{1}{\sin^2 y} \, dy = 9 t \, dt. \] Next, we integrate both sides. The left side can be integrated as follows: \[ \int \frac{1}{\sin^2 y} \, dy = -\cot y + C_1, \] and the right side: \[ \int 9 t \, dt = \frac{9}{2} t^2 + C_2. \] Combining these results, we have: \[ -\cot y = \frac{9}{2} t^2 + C, \] where \( C = C_2 - C_1 \) is a constant. To express this in a more standard implicit form, we can rearrange it: \[ \cot y = -\frac{9}{2} t^2 - C. \] Now, we need to determine the constant \( C \) using the initial condition \( y(1) = \frac{3\pi}{4} \). Calculating \( \cot \left( \frac{3\pi}{4} \right) \): \[ \cot \left( \frac{3\pi}{4} \right) = -1. \] Substituting \( t = 1 \) and \( y = \frac{3\pi}{4} \) into the equation gives: \[ -1 = -\frac{9}{2} (1)^2 - C, \] which simplifies to: \[ -1 = -\frac{9}{2} - C. \] Solving for \( C \): \[ C = -\frac{9}{2} + 1 = -\frac{9}{2} + \frac{2}{2} = -\frac{7}{2}. \] Now substituting \( C \) back into the equation gives: \[ \cot y = -\frac{9}{2} t^2 + \frac{7}{2}. \] Thus, the implicit solution to the differential equation is: \[ \cot y = -\frac{9}{2} t^2 + \frac{7}{2}. \]