Solve struggles to solve equation with variable constraint

I am trying to find {x,y} set satisfying FOC with the constraint:

0

Here is my code, which does not give anything.

Dbuy = 1 - (x/y) - x*Log[y/x];
Dwait = (x/y) - x;
revx = D[(Dbuy*x + Dwait*CCC*y), x]
revy = D[(Dbuy*x + Dwait*CCC*y), y]
revsln=Solve[{revx==0, revy==0},{x,y}]

When I use FindRoot function instead of Solve for a fixed value of CCC (between 0 and 1, for example 0.4), it gives me numerical values (0.322, 0.581) if I add

{{x,0.01}, {y,0.01}} 

But I want some form of closed-form solution for

0<=C<=1

What do you guys think?

Solve does not work

I am trying to find {x,y} set satisfying FOC with the constraint:

0

Here is my code, which does not give anything.

Dbuy = 1 - (x/y) - x*Log[y/x];
Dwait = (x/y) - x;
revx = D[(Dbuy*x + Dwait*CCC*y), x]
revy = D[(Dbuy*x + Dwait*CCC*y), y]
revsln=Solve[{revx==0, revy==0},{x,y}]

When I use FindRoot function instead of Solve for a fixed value of CCC (between 0 and 1, for example 0.4), it gives me numerical values (0.322, 0.581) which work. But I want some form of closed-form solution for

0<=C<=1

What do you guys think?

How do I solve this convolution with a dirac delta?

I am asked to show $$int_{-infty}^{infty}(t-tau)^2delta(tau)dtau = t^2$$.

I proceed by parts: $$int_a^b uv’ = uv|_a^b – int_a^b vu’$$.

I let $v’ = delta(tau)$ and $u = (t-tau)^2$. Then $v = H(tau), u’ = -2(t-tau)$

Then $$int_{-infty}^{infty}(t-tau)^2delta(tau)dtau = (t-tau)^2H(tau)|_{-infty}^infty + int_{infty}^infty H(tau)2(t-tau)dtau$$

How am I expected to solve this given the bounds? Wolfram claims the assertion is true – that it equals $t^2$. How do I go about showing it formally?

Solve system of congruences which involves a quadratic term

I am studying for an admission exam and I came to this system of congruences

$$x^2 equiv 2 text{ mod } 7 hspace{1cm} x equiv 1 text{ mod } 5 $$

I know how to solve linear systems, but I don’t know what to do with a quadratic one.

Could anyone explain me ho to solve it?

Thanks