I need to solve for $x$ in this equation: $$\frac{d^2x}{dy^2} = \frac{1}{2n^2_i\sin^2(\theta_i)} \frac{dn^2}{dx}$$ where $n^2 = a + bx$, with $a, b$ constants and initial values $x = x_i$, $n = n_i$.
My first try is first to take $\dfrac{dn^2}{dx} = \dfrac{d}{dx}(a + bx) = b$,
then $\dfrac{d^2x}{dy^2} = \dfrac{1}{2n^2_i\sin^2(\theta_i)} \dfrac{dn^2}{dx} = d\,\dfrac{d^2x}{dy^2} = \dfrac{b}{2n^2_i\sin^2(\theta_i)}$.
After this I integrate and get $x = \dfrac{by^2}{4n^2_i\sin^2(\theta_i)} + yc_1 + c_2;$ is this correct?
What is the importance of the initial values in this problem?
What type of differential equation is this?
Do you suggest another method that I should try? Thanks!