The Laplace transform turns a linear constant-coefficient initial-value problem into an algebraic equation in the transform variable $s$.

Because transformed derivatives contain the initial values explicitly, the initial conditions enter the algebraic equation directly rather than being imposed later through integration constants.

The procedure is:

1. transform every term in the differential equation;
2. insert the initial values appearing in the transformed derivatives;
3. solve algebraically for the transform $Y(s)$ of the unknown function;
4. use partial fractions or known transform pairs to compute the inverse Laplace transform.

This method is especially convenient for equations with piecewise or switched forcing, and for problems where keeping the initial conditions inside one algebraic calculation is simpler than solving first and fitting constants afterward.

The method does not replace the differential equation: it changes representation so differentiation becomes algebra before the final result is transformed back to time.