# Answer to Question #20653 in Differential Equations for Aqib Meer

Question #20653

use the laplace transform to solve the follwing initial value problem

y''-y'-6y=-6t+11 , y(0)=-1 y'(0)=4

y''-y'-6y=-6t+11 , y(0)=-1 y'(0)=4

Expert's answer

let us write the equation in Laplace domain

(s^2*Y(s) - s*(-1) - 4) - (s*Y(s) - (-1)) - Y(s) = -1!*6/(s^2) + 11/s

Y(s) (s^2-s-1) = -s+3 +11/s-6/s^2

from this we can find Y(s) -& Laplace transform of the function

Y(s) =& (-s+3 +11/s-6/s^2)/(s^2-s-1)

now we can find the function it self, doing the inverse Laplace transform

the result is

-17 + 6*t + 4exp((1/2)*t) * [4cosh((1/2)*t*sqrt(5)) - (6/5)*sqrt(5)*sinh((1/2)*t*sqrt(5))]

(s^2*Y(s) - s*(-1) - 4) - (s*Y(s) - (-1)) - Y(s) = -1!*6/(s^2) + 11/s

Y(s) (s^2-s-1) = -s+3 +11/s-6/s^2

from this we can find Y(s) -& Laplace transform of the function

Y(s) =& (-s+3 +11/s-6/s^2)/(s^2-s-1)

now we can find the function it self, doing the inverse Laplace transform

the result is

-17 + 6*t + 4exp((1/2)*t) * [4cosh((1/2)*t*sqrt(5)) - (6/5)*sqrt(5)*sinh((1/2)*t*sqrt(5))]

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