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 \(t\).
8 Find the particular solution of the differential equation
\(9 \frac{\mathrm{~d}^{2} x}{\mathrm{~d} t^{2}}+6 \frac{\mathrm{~d} x}{\mathrm{~d} t}+x=50 \sin t\)
given that when \(t=0, x=0\) and \(\frac{\mathrm{d} x}{\mathrm{~d} t}=0\).
(i) Find the particular solution of the differential equation
\(\frac{d^2x}{dt^2}+2\frac{dx}{dt}+10x=37\sin3t,\)
given that \(x=3\) and \(\dfrac{dx}{dt}=0\) when \(t=0\).
(ii) Show that, for large positive values of \(t\) and for any initial conditions,
\(x\approx\sqrt{37}\sin(3t-\phi),\)
where \(\phi\) is such that \(\tan\phi=6\).
Find the particular solution of the differential equation
\(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+\frac{\mathrm{d} x}{\mathrm{~d} t}-2 x=2 t^{2}+t-1\)
given that, when \(t=0, x=\frac{\mathrm{d} x}{\mathrm{~d} t}=0\).
Find the particular solution of the differential equation
\(6 \frac{\mathrm{~d}^{2} x}{\mathrm{~d} t^{2}}+3 \frac{\mathrm{~d} x}{\mathrm{~d} t}+6 x=\mathrm{e}^{-t}\)
given that, when \(t=0, x=\frac{\mathrm{d} x}{\mathrm{~d} t}=0\).
(a) Find the general solution of the differential equation
\(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+10 \frac{\mathrm{~d} x}{\mathrm{~d} t}+25 x=338 \sin t\)
(b) Show that, for large positive values of \(t\) and for any initial conditions,
\(x \approx R \sin (t-\phi)\)
where the constants \(R\) and \(\phi\) are to be determined.
It is given that
\(x=\sin ^{-1} t \quad \text { and } \quad y=t \cos ^{-1} t, \quad \text { for } 0 \leqslant t\lt 1 .\)
(a) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-t+\sqrt{1-t^{2}} \cos ^{-1} t\).
(b) Find \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) in terms of \(t\).
The variables \(x\) and \(y\) are related by the differential equation
\(6 \frac{\mathrm{~d}^{2} x}{\mathrm{~d} t^{2}}+5 \frac{\mathrm{~d} x}{\mathrm{~d} t}+x=t^{2}+10 t+13\)
(a) Find the general solution for \(x\) in terms of \(t\).
(b) State an approximate solution for large positive values of \(t\).
Find the particular solution of the differential equation
\(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-12 \frac{\mathrm{~d} x}{\mathrm{~d} t}+36 x=37 \sin t\)
given that, when \(t=0, x=\frac{\mathrm{d} x}{\mathrm{~d} t}=0\).
Find the particular solution of the differential equation
\(3 \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}+2 \frac{\mathrm{~d} y}{\mathrm{~d} x}+y=x^{2},\)
given that, when \(x=0, y=\frac{\mathrm{d} y}{\mathrm{~d} x}=0\).