Solving the Differential Equation Using Laplace Transform

[eddiej90]

Homework Statement

Using the laplace transform technique, solve the ordinary differential equation:
d2y/dt2+4.2dy/dt+4.5y=0 initial conditions: y(0)=1 and y'(0)=1

Homework Equations

From laplace tables:
d2y/dt2=s²Y-sy(0)-y'(0)
dy/dt=sY-y(0)


3. The Attempt at a Solution
After using the laplace tables to convert the equation, and rearranging it to make Y the subject, I have come to this point:
Y=(s+5.2)/s²+4.2s+4.5
Iv tried completing the square on the denominator and other methods but cannot get to a point that looks similar to the laplace tables in order to convert it back. I am pretty sure i need to use the ones under the section "General approach to quadratic functions of the form as2 + bs + c" however i have completely no idea as to what to do with α and β

 

[sunjin09]

http://en.wikipedia.org/wiki/Partial_fraction_decomposition

 

[eddiej90]

Thamkyou for the link, however you cannot factorise the denominator, unless you complete the square. Can you use the partial fraction method with completing the square?

 

[sunjin09]

Thamkyou for the link, however you cannot factorise the denominator, unless you complete the square. Can you use the partial fraction method with completing the square?

Your denominator can certainly complete the square, s^2+4.2s+4.5=(s+2.1)^2-2.1^2+4.5, doesn't it? Obviously no need for partial fraction decomposition