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

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) ={f(t)}

and

n = 1, 2, 3,,

then{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) ={y(t)}.

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

y(t) =⁻¹{Y(s)}.

ty” − y’ = 5t², y(0) = 0

y(t)=

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