Particular solution with initial conditions

Solve the differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} - 7\frac{\mathrm{d}y}{\mathrm{d}x} + 10y = 0,$$ where \(\frac{\mathrm{d}y}{\mathrm{d}x} = 1\) and \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = 8\) when \(x = 0\). [10]

Find the solution of the differential equation $$3\frac{d^2y}{dx^2} + 5\frac{dy}{dx} - 2y = 8 + 6x - 2x^2,$$ where \(y = 6\) and \(\frac{dy}{dx} = 5\) when \(x = 0\). [12]

The following simultaneous equations are to be solved. $$\frac{\mathrm{d}x}{\mathrm{d}t} = 4x + 2y + 6e^{3t}$$ $$\frac{\mathrm{d}y}{\mathrm{d}t} = 6x + 8y + 15e^{3t}$$

1. Show that \(\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} - 12\frac{\mathrm{d}x}{\mathrm{d}t} + 20x = 0\). [5]
2. Given that \(\frac{\mathrm{d}x}{\mathrm{d}t} = 9\) and \(\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = 10\) when \(t = 0\), find the particular solution for \(x\) in terms of \(t\). [7]