91Ó°ÊÓ

Use the method of variation of parameters to find a particular solution of the given differential equation. Then check your answer by using the method of undetermined coefficients. $$ y^{\prime \prime}+2 y^{\prime}+y=3 e^{-t} $$

Write the given expression as a product of two trigonometric functions of different frequencies. \(\cos \pi t+\cos 2 \pi t\)

find the Wronskian of the given pair of functions. $$ e^{-2 t}, \quad t e^{-2 t} $$

In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. \(f(t)=e^{\lambda t} \cos \mu t, \quad g(t)=e^{\lambda t} \sin \mu t, \quad \mu \neq 0\)

Use the method of variation of parameters to find a particular solution of the given differential equation. Then check your answer by using the method of undetermined coefficients. $$ 4 y^{\prime \prime}-4 y^{\prime}+y=16 e^{t / 2} $$

use Euler’s formula to write the given expression in the form a + ib. $$ e^{2-(x / 2) i} $$

Find the general solution of the given differential equation. $$ 2 y^{\prime \prime}-3 y^{\prime}+y=0 $$

find the Wronskian of the given pair of functions. $$ x, \quad x e^{x} $$

In each of Problems 1 through 10 find the general solution of the given differential equation. \(4 y^{\prime \prime}+12 y^{\prime}+9 y=0\)

determine \(\omega_{0}, R,\) and \(\delta\) so as to write the given expression in the form \(u=R \cos \left(\omega_{0} t-\delta\right)\) $$ u=-2 \cos \pi t-3 \sin \pi t $$