In some instances the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. Use Theorem 7.4.1.

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In some instances the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. Use Theorem 7.4.1.

THEOREM 7.4.1 Derivatives of Transforms
If

F(s) =script L{f(t)}

and

n = 1, 2, 3,ldots.gif,

thenscript L{tnf(t)} = (−1)n

dn
dsn

F(s).

Reduce the given differential equation to a linear first-order DE in the transformed function

Y(s) =script L{y(t)}.

Solve the first-order DE for Y(s) and then find

y(t) =script L−1{Y(s)}.

ty” − y’ = 5t2y(0) = 0

y(t)=

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