Solution of Differential Equations with Laplace Transforms

Solution of Differential Equations with Laplace Transforms 

Some differential equations can be solved by taking the Laplace transform of the equation, applying some of the theorems presented in Section 6.5 to obtain an expression for the Laplace transform of the unknown function, and then finding 

We introduce the notation z for the second derivative d z/dfi and z for the first derivative dz/dt. We take the Laplace transform of this equation, applying Eq. (6.72) and the n = 2 version of Eq. (6.73), to express the Laplace transforms of the first and second derivatives. We let Z be the Laplace transform of z. 

When we find the inverse transform of this fimction, we will have our answer. We must carry out some algebraic manipulations before we can find the inverse transforms in Table 6.1. In order to match an expression for a transform in Table 6.1, we complete the square in the denominator (that is, we add a term so that we have a perfect square plus another term)  

We have also expressed the numerator in terms of the quantity that is squared in the denominator. We now make the substitutions. 

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Section 8.7 Solution of Differential Equations with Laplace Transforms... 

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