91Ó°ÊÓ

Find the particular solution of $$ \frac{\mathrm{d} x}{\mathrm{~d} t}=t^{2} $$ that satisfies the condition \(x(3)=0\).

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

Use five steps of Euler's method to find an approximate solution of the initial value problem \(\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{x+x^{2}}{t}, x(1)=-5\) using \(h=0.01\). Work throughout to six decimal places. Hence approximate \(x(1.05)\).

Find the auxiliary equation for the differential equation $$ L \frac{\mathrm{d}^{2} i}{\mathrm{~d} t^{2}}+R \frac{\mathrm{d} i}{\mathrm{~d} t}+\frac{1}{C} i=0 $$ Hence write down the complementary function.

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 $$ y(0)=1, \quad \frac{\mathrm{d} y}{\mathrm{~d} x}(0)=0 $$

Find a first-order equation satisfied by \(x=A \mathrm{e}^{-2 t}\).

Find the general solution of the equation $$ \frac{\mathrm{d} y}{\mathrm{~d} t}+(\tan t) y=\cos t $$

Find the complementary function of the equation $$ \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+\frac{\mathrm{d} y}{\mathrm{~d} x}+y=0 $$

Find the general solution of the following equations: (a) \(\frac{\mathrm{d} y}{\mathrm{~d} x}=3\) (b) \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6 \sin x}{y}\)

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