How do you solve this exchange economy problem using lagrange?

Consider an exchange economy with two consumers (1 and 2) and two goods (X and Y). The preferences of both consumers are represented by a utility function $$U(X,Y)=\sqrt{X}+2\sqrt{Y}$$. Consumer 1 initially has $$X_1^e$$ units of good X and $$Y_1^e$$ units of good Y, and consumer 2 initially has $$X_2^e$$ units of good X and $$Y_2^e$$ units of good Y. The two consumers can trade X at the price of 1 and Y at the price of $$p$$. a) Fully describe the competitive equilibrium in this exchange economy, both as a function of the initial endowments (i.e., $$X^e$$ and $$Y^e$$) and for the specific case of $$(X^e, Y^e) = (6, 1)$$, $$(X_2^e, Y_2^e) = (10, 3)$$.

I tried to do $$L = \sqrt{X} +2\sqrt{Y} + \lambda(X_1^e+ p Y_1^e - X_1^c-p Y_1^c)$$. but it definetly did not work.

Given a pure-exchange economy:

$$u_i(x_i,y_i)=\sqrt{x_i}+2\sqrt{y_i}$$ for $$i\in\{1,2\}$$

$$\omega_1=(\omega_1^X,\omega_1^Y)=(6,1)$$ and $$\omega_2=(\omega_2^X,\omega_2^Y)=(10,3)$$

we want to find the competitive equilibrium price $$p > 0$$, where price of $$X$$ is $$1$$ and price of $$Y$$ is $$p$$.

To do so, first we solve the utility maximisation problem of each consumer $$i$$.

$$\begin{eqnarray*} \max_{x_i\geq 0, \ y_i\geq 0} & \sqrt{x_i}+2\sqrt{y_i} \\ \text{s.t. } & x_i + py_i \leq \omega_i^X+p\omega_i^Y\end{eqnarray*}$$

Setting up the Lagrangian, we get

$$\mathcal{L} = \sqrt{x_i}+2\sqrt{y_i} - \lambda(x_i + py_i - \omega_i^X-p\omega_i^Y)$$

First-order conditions are: $$\begin{eqnarray*} \dfrac{\partial \mathcal{L}}{\partial \mathcal{x_i}} & = & \dfrac{1}{2\sqrt{x_i}}-\lambda= 0\\ \dfrac{\partial\mathcal{L}}{\partial \mathcal{y_i}} & = & \dfrac{1}{\sqrt{y_i}}-\lambda p= 0 \\ \dfrac{\partial\mathcal{L}}{\partial \lambda } & = & x_i + py_i - \omega_i^X-p\omega_i^Y = 0\end{eqnarray*}$$

Eliminating $$\lambda$$, we observe that $$x_i, y_i$$ satisfy:

$$y_i=\dfrac{4x_i}{p^2}$$ and

$$x_i + py_i = \omega_i^X+p\omega_i^Y$$

Solving for $$x_i$$, $$y_i$$, we get $$x_i= \dfrac{p(\omega_i^X+p\omega_i^Y)}{p+4}$$ and $$y_i= \dfrac{4(\omega_i^X+p\omega_i^Y)}{p(p+4)}$$.

To solve for competitive equilibrium price, we can consider market for $$X$$ and set total demand for $$X$$ equal to total supply of $$X$$, and we get the following condition $$\dfrac{p(\omega_1^X+p\omega_1^Y)}{p+4}+ \dfrac{p(\omega_2^X+p\omega_2^Y)}{p+4} =\omega_1^X+\omega_2^X$$

Equilibrium $$p$$ solves the above condition and solving it we get: $$p=2\sqrt{\dfrac{\omega_1^X+\omega_2^X}{\omega_1^Y+\omega_2^Y}}$$

For the specific case when $$\omega_1=(\omega_1^X,\omega_1^Y)=(6,1)$$ and $$\omega_2=(\omega_2^X,\omega_2^Y)=(10,3)$$, we get $$p =4$$. Corresponding competitive equilibrium allocation is $$(x_1,y_1)= \left(5,\frac{5}{4}\right)$$ and $$(x_2,y_2)=\left(11,\frac{11}{4}\right)$$