Differential Equations Answers

 Find the Laplace transform, if it exists, of each of the following functions

(a) f(t) = e^at

(b) f(t) = 1

(c) f(t) = t 

Find the general solution to y'' − y' − 2y = 2e^3x

Find the general solution to y'' + 4y' + 3y = x.

Given that p(x) = −2 is a particular solution to y'' − 3y' − 4y = 8, write the general solution and verify that the general solution satisfies the equation. 

Given that p(x) = x is a particular solution to the differential equation y'' + y = x write the generalized solution and check by verifying that the solution satisfies the equation.

Verify that the given functions form the fundamental set of solution of the

differential equation on the given indicated interval.

y"-2y' + 5y = 0

y= c₁e^x cos 2x + c2₂e^x sin 2x, (-infinity, +infinity)

Use the method of variation of parameters to solve the system

𝑋′ =[

3 −1 −1

−2 3 2

4 −1 −2

]𝑋+[

1

𝑒𝑡

𝑒𝑡]