91Ó°ÊÓ

Solve \(\frac{\mathrm{d} x}{\mathrm{~d} t}=t-t x, x(0)=0\).

Find the general solutions of the following equations: (a) \(\frac{\mathrm{d} x}{\mathrm{~d} t}=x t\) (b) \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x}{y}\) (c) \(t \frac{\mathrm{d} x}{\mathrm{~d} t}=\tan x\) (d) \(\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{x^{2}-1}{t}\)

Find the general solution of \(y^{\prime \prime}+16 y=x^{2}\)

Use an integrating factor to obtain the general solution of \(i R+L \frac{\mathrm{d} i}{\mathrm{~d} t}=\sin \omega t\), where \(R, L\) and \(\omega\) are constants.

Find the general solution of the equation \(\frac{\mathrm{d} x}{\mathrm{~d} t}=t(x-2)\). Find the particular solution which satisfies \(x(0)=5\)

Find the particular solution of \(y^{\prime \prime}+3 y^{\prime}-4 y=\mathrm{e}^{x}, y(0)=2, y^{\prime}(0)=0\)

Identify the dependent and independent variables of the following differential equations. Give the order of the equations and state which are linear. (a) \(\frac{\mathrm{d} y}{\mathrm{~d} x}+9 y=0\) (b) \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)\left(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right)+3 \frac{\mathrm{d} y}{\mathrm{~d} x}=0\) (c) \(\frac{\mathrm{d}^{3} x}{\mathrm{~d} t^{3}}+5 \frac{\mathrm{d} x}{\mathrm{~d} t}=\sin x\)

Find a particular integral for the equation $$ \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+\frac{\mathrm{d} y}{\mathrm{~d} x}+y=1+x $$

Show that \(x(t)=7 \cos 3 t-2 \sin 2 t\) is a solution of \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+2 x=-49 \cos 3 t+4 \sin 2 t\)

Use an integrating factor to solve the differential equation $$ \frac{\mathrm{d} x}{\mathrm{~d} t}+x \cot t=\cos 3 t $$