Differential Equations

Laplace Transforms Exercise 1 : Applying Laplace Transforms Obtain the Laplace transforms for the following functions: $\frac{1}{4}t^3$ $5e^{-2t}$ $4\cos 3t$ $\sin^2 t$ Exercise 2 : Inverse Laplace Transforms Obtain the inverse transforms for the following: $\frac{1}{4s^2 + 1}$ $\frac{1}{s(s + 4)}$ $\frac{2}{s(s^2 + 9)}$ $\frac{6}{1 - s^2}$ $\frac{1}{s^2(s^2 + 1)}$ $\frac{4}{s(s^2 - 6s + 8)}$ Exercise 3 : Solving Linear ODEs via Laplace Solve the following linear differential equations with constant coefficients: ...

Differential Equations Exercise 1 : Classification — Linear vs Nonlinear Which of the following are linear first- and second-order DEs with constant coefficients? $y’ + x^2 y = 2x$ $5y’’ - 2y’ - 4x = 3y$ $y^4 + 2y’’ + 3y’ = 0$ $\sin x, y’’ - y = 0$ $y’’ - x^5 = 2$ $2y’’ - y’ + \frac{3}{2}y = 0$ Exercise 2 : Homogeneity and Order Which of the following are homogeneous and non-homogeneous DEs and what is the order in each case? ...