7 Find the particular solution of the differential equation
\(10 \frac{\mathrm{~d}^{2} x}{\mathrm{~d} t^{2}}+3 \frac{\mathrm{~d} x}{\mathrm{~d} t}-x=t+2\)
given that when \(t=0, x=0\) and \(\frac{\mathrm{d} x}{\mathrm{~d} t}=0\).

Find the solution of the differential equation
\(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+6 \frac{\mathrm{~d} x}{\mathrm{~d} t}+9 x=18 t^{2}+6 t+1\)
given that, when \(t=0, x=3\) and \(\frac{\mathrm{d} x}{\mathrm{~d} t}=0\).

Find the particular solution of the differential equation
\(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+0.16 \frac{\mathrm{~d} x}{\mathrm{~d} t}+0.0064 x=8.64+0.32 t\)
given that when \(t=0, x=0\) and \(\frac{\mathrm{d} x}{\mathrm{~d} t}=0\).

Show that, for large positive \(t, \frac{\mathrm{~d} x}{\mathrm{~d} t} \approx 50\).

Find the general solution of the differential equation
\(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+4 \frac{\mathrm{~d} x}{\mathrm{~d} t}+4 x=7-2 t^{2}\)

Find the general solution of the differential equation
\(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+2 \frac{\mathrm{~d} x}{\mathrm{~d} t}+5 x=4-5 t^{2}\)

Given that
\(x \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}+(2 x+2) \frac{\mathrm{d} y}{\mathrm{~d} x}+(2-3 x) y=10 \mathrm{e}^{2 x}\)
and that \(v=x y\), show that
\(\frac{\mathrm{d}^{2} v}{\mathrm{~d} x^{2}}+2 \frac{\mathrm{~d} v}{\mathrm{~d} x}-3 v=10 \mathrm{e}^{2 x}\)

Find the general solution for \(y\) in terms of \(x\).

Given that
\(x \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}+(2 x+2) \frac{\mathrm{d} y}{\mathrm{~d} x}+(2-3 x) y=10 \mathrm{e}^{2 x}\)
and that \(v=x y\), show that
\(\frac{\mathrm{d}^{2} v}{\mathrm{~d} x^{2}}+2 \frac{\mathrm{~d} v}{\mathrm{~d} x}-3 v=10 \mathrm{e}^{2 x}\)

Find the general solution for \(y\) in terms of \(x\).

Find \(x\) in terms of \(t\) given that
\(4 \frac{\mathrm{~d}^{2} x}{\mathrm{~d} t^{2}}+4 \frac{\mathrm{~d} x}{\mathrm{~d} t}+x=6 \mathrm{e}^{-2 t}\)
and that, when \(t=0, x=\frac{5}{3}\) and \(\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{7}{6}\).

State \(\lim _{t \rightarrow \infty} x\).
[Questions 10 and 11 are printed on the next page.]

Find the general solution of the differential equation
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+2 \frac{\mathrm{~d} y}{\mathrm{~d} x}+4 y=4 x^{2}+8\)

Find \(y\) in terms of \(t\), given that
\(5 \frac{\mathrm{~d}^{2} y}{\mathrm{~d} t^{2}}+6 \frac{\mathrm{~d} y}{\mathrm{~d} t}+5 y=15+12 t+5 t^{2}\)
and that \(y=\frac{\mathrm{d} y}{\mathrm{~d} t}=0\) when \(t=0\).

Find the general solution of the differential equation
\(4 \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}+4 \frac{\mathrm{~d} y}{\mathrm{~d} x}+65 y=65 x^{2}+8 x+73 .\)

Show that, whatever the initial conditions, \(\frac{y}{x^{2}} \rightarrow 1\) as \(x \rightarrow \infty\).

It is given that
\(x=t^{2} \mathrm{e}^{-t^{2}} \quad \text { and } \quad y=t \mathrm{e}^{-t^{2}}\)
(i) Show that
\(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1-2 t^{2}}{2 t-2 t^{3}} .\)
(ii) Find \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) in terms of \(t\).