I have the following differential equation. Basically, I want to verify that the following steps I've taken are kosher. I'd appreciate any feedback:
$$\begin{split}0 = &-y\int_{0}^{y}\int_{0}^{y}zxF'(x)F'(z)dxdz + \frac{y-1}{3}\int_{0}^{1}\int_{0}^{1}(zx-x-z-2)F'(x)F'(z)dxdz\\ &+ y\int_{0}^{1}\int_{0}^{y}xF'(x)F'(z)dxdz + y\int_{0}^{y}\int_{0}^{1}zF'(x)F'(z)dxdz\end{split}$$
where $\int_{0}^{1}F'(x)dx= (1-a)$. Thus, I can rewrite this as
$$\begin{split}0 = &-y\bigg(\int_{0}^{y}xF'(x)dx\bigg)^{2} + \frac{y-1}{3}\int_{0}^{1}\int_{0}^{1}(zx-x-z-2)F'(x)F'(z)dxdz\\ &+ 2(1-a)y\int_{0}^{y}xF'(x)dx\end{split}$$
I take the derivative of both sides (omitting the LHS, since it's just $0$)
$$\begin{split} &- \bigg(\int_{0}^{y}xF'(x)dx\bigg)^{2} - 2y^2F'(y)\bigg(\int_{0}^{y}xF'(x)dx\bigg)\\ &+ \frac{1}{3}\int_{0}^{1}\int_{0}^{1}(zx-x-z-2)F'(x)F'(z)dxdz + 2(1-a)\int_{0}^{y}xF'(x)dx + 2(1-a)y^2F'(y)\end{split}$$
Taking the derivative again,
$$\begin{split}0 = &- 6yF'(y)\bigg(\int_{0}^{y}xF'(x)dx\bigg) - 2y^2F''(y)\bigg(\int_{0}^{y}xF'(x)dx\bigg) - 2y^3(F'(y))^{2}\\ &+ 6(1-a)yF'(y) + 2(1-a)y^2F''(y)\end{split}$$
Or,
$$\begin{split} \bigg(6yF'(y) + 2y^2F''(y)\bigg)\bigg(\int_{0}^{y}xF'(x)dx\bigg) &= - 2y^3(F'(y))^{2} + (1-a)\bigg(6yF'(y) + 2y^2F''(y)\bigg)\end{split}$$
Or,
$$\begin{split} \int_{0}^{y}xF'(x)dx &= - \frac{y^2(F'(y))^{2}}{3F'(y) + yF''(y)} + (1-a)\end{split}$$
Differentiating both sides again,
$$\begin{split} yF'(y) &= - \frac{2y(F'(y))^{2} + 4y^2F'(y)F''(y)}{3F'(y) + yF''(y)} - \frac{\big(y^2(F'(y))^{2}\big)\big(3F''(y) + F''(y) + yF'''(y)\big)}{\big(3F'(y) + yF''(y)\big)^{2}}\\ 1 &= - \frac{2F'(y) + 4yF''(y)}{3F'(y) + yF''(y)} - \frac{yF'(y)\big(3F''(y) + F''(y) + yF'''(y)\big)}{\big(3F'(y) + yF''(y)\big)^{2}}\end{split}$$
$$\begin{split} yF'(y) &= - \frac{2y(F'(y))^{2} + \color{red}2y^2F'(y)F''(y)}{3F'(y) + yF''(y)} \color{red}+ \frac{\big(y^2(F'(y))^{2}\big)\big(3F''(y) + F''(y) + yF'''(y)\big)}{\big(3F'(y) + yF''(y)\big)^{2}}\\ 1 &= - \frac{2F'(y) + \color{red}2yF''(y)}{3F'(y) + yF''(y)} \color{red}+ \frac{yF'(y)\big(3F''(y) + F''(y) + yF'''(y)\big)}{\big(3F'(y) + yF''(y)\big)^{2}}\end{split}$$
Also, remember to assume/ check the condition when you apply fundamental theorem of calculus.