Solving Ordinary Differential Equations

ODEs Can be solved algebraically by using Laplace transform.

Steps

1. Laplace transform each term using derivative rules.
2. Substitute initial conditions directly.
3. Solve algebraic equation in $s$-domain.
4. Apply inverse Laplace transform.

The above steps can be applied for a system of ODEs as well.

Real-World Applications

LCR Circuits

Differential equation:

$$L \frac{di}{dt} + Ri(t) + \frac{1}{C}\int_0^t i(x)dx = v(t)$$

Laplace converts it into an algebraic relation involving ( I(s) ) and ( V(s) ).

LTI Systems

Impulse response $h(t)$, transfer function $H(s) = L{h(t)}$.