Question

Use Laplace transforms to solve the following initial value problems.

(a) Y '′ +9y=0,y(0)=3,y'(0)=12
(b) Y −7y=te ⁷ᵗ,y(0)=0

Answer 1

The student is asked to solve two distinct initial value problems using Laplace transforms: a second-order homogeneous and a first-order non-homogeneous differential equation, applying relevant properties and theorems to find the solutions.

Step-by-step explanation:

In mathematics, specifically in the field of differential equations, the student is asked to solve two separate initial value problems using Laplace transforms. In these problems, we are given differential equations and the initial conditions, which allow us to uniquely determine the solution to the problems. The two initial value problems are:

• (a) Y '′ +9y=0, y(0)=3, y'(0)=12: This is a second-order linear homogeneous differential equation with constant coefficients.
• (b) Y −7y=te⁷ᵗ, y(0)=0: This is a first-order linear non-homogeneous differential equation.

For each problem, one would take the Laplace transform of both sides of the given differential equation, use the initial conditions to solve for the transform of y, and then find the inverse Laplace transform to obtain the solution in the time domain. It's essential to correctly apply properties of Laplace transforms and the initial value theorem to solve for the unknowns in the transformed domain.