91Ó°ÊÓ

Find a particular integral for the equation $$ \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-x=4 \mathrm{e}^{-2 t} $$

Find the general solution of \(\frac{\mathrm{d} x}{\mathrm{~d} t}=2 x+4 t\). What is the particular solution which satisfies \(x(1)=2 ?\)

Find the general solution of \(\frac{\mathrm{d} y}{\mathrm{~d} x}+y=2 x+5\).

Solve \(\frac{\mathrm{d} x}{\mathrm{~d} t}+2 x=\mathrm{e}^{2 t} \cos t\) (a) by using an integrating factor (b) by finding its complementary function and a particular integral.

Obtain the first four non-zero terms in the power series solution of the initial value problem \(\frac{\mathrm{d} y}{\mathrm{~d} x}+x y=0, y(0)=1\)

Find the general solutions of the following equations: (a) \(\frac{\mathrm{d} y}{\mathrm{~d} x}=k x, \quad k\) constant (b) \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-k y, \quad k\) constant (c) \(\frac{\mathrm{d} y}{\mathrm{~d} x}=y^{2}\) (d) \(y \frac{\mathrm{d} y}{\mathrm{~d} x}=\sin x\) (e) \(y \frac{\mathrm{d} y}{\mathrm{~d} x}=x+2\) (f) \(x^{2} \frac{\mathrm{d} y}{\mathrm{~d} x}=2 y^{2}+y x\) (g) \(\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{t^{4}}{x^{5}}\)

Verify that \(y=A \cos x+B \sin x\) satisfies the differential equation. $$ \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+y=0 $$ Verify also that \(y=A \cos x\) and \(y=B \sin x\) each individually satisfy the equation.

Obtain the general solution of \(y^{\prime \prime}-y^{\prime}-2 y=6\)

Obtain the power series solution of the equation \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+x y=0\), up to terms involving \(x^{7}\). This differential equation is known as an Airy equation.

Obtain the general solution of the equation $$ \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+3 \frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=10 \cos 2 x $$ Find the particular solution satisfying $$ y(0)=1, \frac{\mathrm{d} y}{\mathrm{~d} x}(0)=0 $$