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Magazine Problem 20
Use the Laplace transform to solve the second-order initial value problems in Exercises 11-26. $$ y^{\prime \prime}-2 y^{\prime}-3 y=e^{4 t}, \quad y(0)=1, \quad y^{\prime}(0)=-1 $$
Problem 20
Find the inverse Laplace transform of each function in Exercises 16-25. Create a piecewise definition for your solution that doesn't use the Heaviside function. $$ F(s)=\frac{e^{-s}}{s^{2}+4} $$
Problem 21
Use the Laplace transform to solve the second-order initial value problems in Exercises 11-26. $$ y^{\prime \prime}-y^{\prime}-2 y=e^{2 t}, \quad y(0)=-1, \quad y^{\prime}(0)=0 $$
Problem 21
Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{2 s-1}{(s+1)(s-2)} $$
Problem 21
Find the inverse Laplace transform of each function in Exercises 16-25. Create a piecewise definition for your solution that doesn't use the Heaviside function. $$ F(s)=\frac{s e^{-3 s}}{s^{2}+4} $$
Problem 22
Use the Laplace transform to solve the second-order initial value problems in Exercises 11-26. $$ y^{\prime \prime}-3 y^{\prime}-4 y=e^{-t}, \quad y(0)=0, \quad y^{\prime}(0)=-1 $$
Problem 22
Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{2 s-2}{(s-4)(s+2)} $$
Problem 23
Use the Laplace transform to solve the second-order initial value problems in Exercises 11-26. $$ y^{\prime \prime}+2 y^{\prime}+2 y=\cos t, \quad y(0)=-1, \quad y^{\prime}(0)=0 $$
Problem 23
Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{7 s+13}{s^{2}+2 s-3} $$
Problem 24
Perform the appropriate partial fraction decomposition, and then use the result to find the inverse Laplace transform of the given function. $$ Y(s)=\frac{7-s}{s^{2}+s-2} $$
