Solving a Partially Decoupled System with Initial Values

[cheddahchad]

Homework Statement

dx/dt = 2x - 3y²
dy/dt = -3y

Derive the general solution and find the solution that satisfies the initial values: x(0) = 0 and y(0) = 1.

The attempt at a solution

dy/dt = -3y
y(t) = c₁e^-3t

dx/dt = 2x - 3(c₁e^-3t)²

I have no idea where to go from here. Any help?

 

[hilbert2]

The equation $\frac{dx}{dt}-2x=-3(c_{1}e^{-3t})^{2}$ is just a linear, inhomogenous 1st order DE. The method of solving it is multiplying both sides of the eq with the integrating factor $e^{-2t}$ and using the "derivative of product" rule to get $\frac{d}{dt}(e^{-2t}x)=-3e^{-2t}(c_{1}e^{-3t})^{2}$. This equation can be directly integrated to find the function x(t).