Practice question · Put in order

Order the steps of solving an initial-value problem with the Laplace transform.

Solve the resulting algebraic equation for $Y(s)$
Split $Y(s)$ into partial fractions
Transform both sides, turning derivatives into multiplications by $s$
Insert the initial conditions the transform absorbed
Invert each simple piece back to the time domain

Hints

The whole point of the method: trade calculus in $t$ for algebra in $s$ .
Initial conditions enter immediately via $\mathcal{L}\{y'\} = sY - y(0)$ , not at the end as integration constants.

Show the answer

Transform both sides, turning derivatives into multiplications by $s$
Insert the initial conditions the transform absorbed
Solve the resulting algebraic equation for $Y(s)$
Split $Y(s)$ into partial fractions
Invert each simple piece back to the time domain

Why

Laplace converts an ODE into algebra, bakes the initial conditions in from the start, and a table lookup brings the answer home. Its real superpower: discontinuous inputs (switches, impulses) that defeat classical methods are handled effortlessly.

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Order the steps of solving an initial-value problem with the Laplace transform.

Practise Laplace Transforms

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