I am trying to predict the covariance of two linear combinations of normal random variables:
$\newcommand{\N}{\mathcal N}$
\begin{align}
X &= w\N(u_1,\sigma^2_1)+(1-w)\N(u_2,\sigma^2_2)  \\
Y &= w\N(u_1,\sigma^2_1)+(1-w)\N(u_3,\sigma^2_3)
\end{align}
where $w$ can range from $0$ to $1$.
 
I've tried solving for $\text{cov}(X,Y)$ using
\begin{align}
\text{cov}(X,Y) &= \text{E}(XY) - \text{E}(X)\text{E}(Y) \\  
\text{cov}(X,Y) &= \text{corr}(X,Y)\sigma_X\sigma_Y
\end{align}

but am not sure how to find $\text{E}(XY)$ in the first case and $\text{corr}(X,Y)$ in the second.