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*) = *ℒ*{*f*(*t*)}

and

*n* = 1, 2, 3, ,

then*ℒ*{*t*^{n}*f*(*t*)} = (−1)^{n}

d^{n} |

ds^{n} |

*F*(*s*).

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

*Y*(*s*) = *ℒ*{*y*(*t*)}.

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

*y*(*t*) = *ℒ*^{−1}{*Y*(*s*)}.

*ty”* − *y’* = 6*t*^{2}, *y*(0) = 0

*y*(*t*) =