91Ó°ÊÓ

determine whether the given set of functions is linearly dependent or linearly independent. If they are linearly dependent, find a linear relation among them. $$ f_{1}(t)=2 t-3, \quad f_{2}(t)=t^{2}+1, \quad f_{3}(t)=2 t^{2}-t $$

Determine the general solution of the given differential equation. \(y^{\mathrm{vi}}+y^{\prime \prime \prime}=t\)

Find the general solution of the given differential equation. Leave your answer in terms of one or more integrals. $$ y^{\prime \prime \prime}-y^{\prime}=\csc t, \quad 0

Follow the procedure illustrated in Example 4 to determine the indicated roots of the given complex number. $$ (1-i)^{1 / 2} $$

determine whether the given set of functions is linearly dependent or linearly independent. If they are linearly dependent, find a linear relation among them. $$ f_{1}(t)=2 t-3, \quad f_{2}(t)=2 t^{2}+1, \quad f_{3}(t)=3 t^{2}+t $$

Follow the procedure illustrated in Example 4 to determine the indicated roots of the given complex number. $$ 1^{1 / 4} $$

determine whether the given set of functions is linearly dependent or linearly independent. If they are linearly dependent, find a linear relation among them. $$ f_{1}(t)=2 t-3, \quad f_{2}(t)=t^{2}+1, \quad f_{3}(t)=2 t^{2}-t, \quad f_{4}(t)=t^{2}+t+1 $$

Find the solution of the given initial value problem. Then plot a graph of the solution. \(y^{\prime \prime \prime}+4 y^{\prime}=t, \quad y(0)=y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=1\)

Follow the procedure illustrated in Example 4 to determine the indicated roots of the given complex number. $$ [2(\cos \pi / 3+i \sin \pi / 3)]^{1 / 2} $$

Find the solution of the given initial value problem. Then plot a graph of the solution. \(y^{\mathrm{iv}}+2 y^{\prime \prime}+y=3 t+4, \quad y(0)=y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=y^{\prime \prime \prime}(0)=1\)