Suppose that the following function is the utility function of a representative consumer: $$U(x_1,x_2,y)=a/(b-c) *(x_1+x_2)-b/(2*(b-c)^2)*(x_1^2+x_2^2)-c/(b^2-c^2)*x_1*x_2+y$$
The budget restriction is given by $p_1*x_1+p_2*x_2+y=M$
$p_i$ represents the prices for good $x_1$ respectively $x_2$. $y$ is the income left for consumption of other goods. $M$ is total income.
The solution of the utility maximization problem is given by the following demand functions for good $x_1$ and $x_2$: $$x_1=a-b*p_1+c*p_2$$ and $$x_2=a-b*p_2+c*p_1$$
I cannot solve for these demand functions (I get very complicated terms instead of these easy demand functions). What I did was: I calculated the following Langrange function: $$L=a/(b-c) *(x_1+x_2)-b/(2*(b-c)^2)*(x_1^2+x_2^2)-c/(b^2-c^2)*x_1*x_2+y+lambda*(M-p_1*x_1-p_2*x_2-y).$$ Then I maximized $L$ with respect to $x_1$, $x_2$ and $y$. The derivative of $U(x_1,x_2,y)$ with respect to $y$ leads to $lamda=1$. I plugged $lambda=1$ into the first order conditions of $x_1$ and $x_2$. However, subsequently solving for $x_1$ and $x_2$ leads to very complicated terms instead of these easy demand functions.
I am very glad for any help!! What is wrong with my approach?