Why should $dp_2=dm =0$ in this problem? I am studying Essential Mathematics for Economic Analysis and in chapter 12 problem 10 there is a problem that I can solve but I do not understand why the solution works.
The problem asks us to differentiate the following system from consumer demand theory:
$$U'_1(x_1,x_2) = \lambda p_1 \\ U'_2(x_1,x_2) = \lambda p_2 \\ p_1 x_1 + p_2x_2=m $$
where system defines $x_1, x_2$ and $\lambda$ as differentiable functions functions of $p_1,p_2$ and $m$. We are asked afterwards find expression for $\partial x_1 / \partial p_1$.
Now I have no problem with the first step - the differentiated system will be:
$$U^{''}_{11}(x_1,x_2)dx_1 + U^{''}_{12}(x_1,x_2)dx_2 = \lambda dp_1 + p_1 d\lambda   \\ U{''}_{21}(x_1,x_2)dx_1 + U{''}_{22}(x_1,x_2)dx_2 = \lambda dp_2 + p_2 d\lambda \\ p_1 dx_1 + x_1 dp_1 + p_2 d x_2 + x_2 dp_2=dm $$
Now my first attempt was to solve 2nd equation for $dx_2$ substitute it into the first and solve for $d x_1 /dp_1$ but I got wrong result. Afterwards in the solution in the back there is a hint that says that to solve this we should put $dp_2=dm=0$ after I follow this hint I get the correct solution:
$$\partial x_1 / \partial p_1 = \frac{[\lambda  p_2^2 + x_1(p_2 U^{''}_{12}-p_1U^{''}_{21})]}{(p_1^2 U^{''}_{22} -2p_1p_2U^{''}_{12} + p_2^2 U^{''}_{11})}$$
But I don't get why we can simply assume $d p_2 = dm = 0$ that seems so arbitrary, how come that  is necessary  to find the correct solution?
The assumption $dm =0$ says that we examine the behavior of the consumer under a fixed nominal income, and this is something interesting to study, because it aligns to a large degree with the observed reality of many people that have approximately constant income.
The assumption $dp_2=0$ assumes away general equilibrium effects, since we are looking at changes in $p_1$ which is price in one market, while $p_2$ is the price in another market, presumably of a different good. So while formally the justification here is that &quot;we do partial equilibrium analysis&quot;, the vindication of partial equilibrium analysis is that the effects from changes in one goods market to another goods market take some time to materialize, so partial equilibrium analysis has relevance to the real world.
So we see that the assumption $dm=dp_2=0$ is not &quot;necessary to find the correct solution&quot; (there is not one correct solution, but solutions each consistent with the corresponding assumptions made). But it is neither an &quot;arbitrary&quot; set of assumptions, on the contrary, it has a specific logic and purpose for which it is imposed.
The OP asked in the comments

&quot;Then why not also assume that $d\lambda=0$&quot;?

But the optimal value of lambda is endogenous to the model, and it summarizes aspects of the consumer's optimal behavior. So whether it changes or stays constant is for the model to tell us.