You used Dx twice instead of Dy .
The solution to your system is not unique. The general solution is
x(t) = arbitrary subject to x(0) = 0
y(t) = (int((1/30*(22*x(t)+3*(diff(x(t), t))))*exp((2/3)*t), t)+C1)*exp(-(2/3)*t)
where C1 is a constant of integration needed to provide the y(0) = 0 . The value of it will depend upon x(t) .
You can use any of a number of functions for x(t). For example, x(t) = t^2 is fine, as is x(t) = cos(t) - 1 or x(t) = exp(t) - 1 .
