Help me find the solution to the IVP in implicit form?
Just a $\color{red}{\mbox{typo}}$, I think, in the first place. The differential equation is an exact one if:
$$
\left[105\sin(3y-15x)-y\right]dx+\left[-21\sin(3y-15\color{red}{x})-x+2y\right]dy=0
$$
So it is supposed that such is the case. When put:
$$
du = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy = 0
$$
then we have:
$$
\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial x}\right) =
\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial y}\right) = 315\cos(3 y - 15 x) - 1
$$
And the original ODE is reproduced with:
$$
u(x,y)=7\cos(3y-15x)-yx+y^2 \quad \Longrightarrow \quad \left\{
\begin{array}{l} \partial u/\partial x = 105\sin(3 y - 15 x) - y \\
\partial u/\partial y = -21\sin(3 y - 15 x) - x + 2 y \end{array} \right.
$$
We know that $du=0$ , so indeed $u(x,y) = C$ is the solution.
However, $C$ can be anything, not just $C=1287$ ; I don't see either how to make sense of the additional "condition" $y(8)=40$ .