Finding a (distribution) $p(\cdot)$ satisfying $p(x) = \int_0^\infty \exp(\lambda_1 y^2 + \lambda_2 x y) p(y) dy$ and $p(x) = 0$ for $x < 0$

Question

As the title says, I'm currently trying to find a distribution $p(\cdot)$ satisfying

1. $p(x) = \int_0^\infty \exp(\lambda_1 y^2 + \lambda_2 x y) p(y) dy$
2. $\forall x < 0 : p(x) = 0$

The only way I have so far thought of trying was to search for multivariate distributions with equal marginals where the conditional distribution $p(x \mid y)$ is of the form $\exp(\lambda_1 y^2 + \lambda_2 x y)$.

Any help on how I can tackle this problem is greatly appreciated!

Answer 1

Hint: $$\int_0^\infty p(x)\exp(-y)\operatorname d y = p(x)$$

So a solution might be some $p(\;)$ such that: $p(x)\exp(-y) =p(y)\exp(y(\lambda_1 y+\lambda_2 x))$