Note: I asked in the Mathematics Meta regarding if it is permitted to ask Mathematics questions of economic nature. I also posted this question in the Economics Stackexchange but have not got any replies regarding the mathematical nature of this problem so I'm going to try it here. Let me know if I did not give enough information Mathematically
I know that the Slutsky equation is defined as:
$\frac{\partial x_1^s}{\partial p_1} = \frac{\partial x_1^m}{\partial p_1} + x_1^o \frac{\partial x_1^m}{\partial m}$
My problem is right now is making use of this information given (I am aware of how to take partial derivatives) but cannot seem to understand how to apply it to problem sets.
Here's an example (I'm more concerned about the steps on how to get to the answer not just the answer);
A consumer has preferences given by $U(x_1,x_2)= x_1^2x_2$
(a) Derive the demand curves for $x_1, x_2$ when prices and income are given by $p_1, p_2$ and $m$
$x_1^*=2m/3p_1$ and $x_2^*=m/3p_2$ -I think I understood how to do that
(b) Illustrate the equilibrium on a diagram when $p_1$ = $p_2$ $ =$ 1 and $I$ = $12
- the way I did this was by graphing and simply finding the equllibrium point graphically based on the Demands for goods $1$ and $2$ on the budget line
(c) Calculate the exact income and substitution effects for $x_1$ when $p_1$ rises to $3.
-The only way I'm currently able to do this is without Calculus, as described in this video which doesn't seem to sit well with me being that the Slutsky Equation is defined very clearly with use of calculus. I just don't know how to apply it.
(d) Explain your exact results using the appropriate Slutsky equation.
- same problem here.
Another Note: I'm no simply looking for someone to "do my homework" I'm primarily interested is in knowing how to apply the Slutsky equation when facing similar problems.
Thanks in advance.