Use the Laplace transform to solve 2y y 0 y0 3 SolutionTh

Use the Laplace transform to solve 2y\' + y = 0 y(0) = -3

Solution

The first step in using Laplace transforms to solve an IVP is to take the transform of every term in the differential equation.

L{2y\'}+L{y}=0

2L{y\'}+L{y}=0

2[sY(s)-y(0)] + Y(s) = 0

2sY(s) +6 +Y(s) = 0

(2s+1)Y(s) = -6

Y(s) = -3 / (2s+1)

Y(s) = -3/2 * 1/(s+1/2)

y(t) = -3/2 * e^-t/2